If the universe is infinite, does that mean that everything exists somewhere?

[Richard87]

If the universe is infinite, does that mean that everything exists somewhere, besides obviously impossible things like a star that contains oxygen but doesn't contain oxygen or a 4-sided triangle?

[mgb_phys]

It would if it was but it isn't

[Chalnoth]

If the universe is infinite, does that mean that everything exists somewhere, besides obviously impossible things like a star that contains oxygen but doesn't contain oxygen or a 4-sided triangle?
Well, consider this by way of analogy.

The set of all even numbers is infinite. I can go on counting even numbers for ever and ever and never reach an end.

But clearly the set of all even numbers does not include all possible numbers. It doesn't include, for instance, the number pi.

So even if the universe is infinite (we don't know whether or not it is), then that doesn't necessarily mean that all possibilities are realized.

However, there may be other reasons to believe that all possibilities are realized, mainly stemming from quantum mechanics, where we find, for instance, that if there is the possibility of matter inhabiting a region of space, then particles of that sort of matter will necessarily pop in and out of the vacuum. Another way of saying this is that in quantum mechanics, there mere possibility of existence forces existence. So it is not unreasonable to suspect that perhaps all possibilities must actually be realized.

This doesn't mean that anything and everything we can imagine occurs, of course. We can imagine quite a lot of impossible things, as you mention above. But we can also imagine a great many things that are not obviously impossible, and yet may turn out to be upon deeper inspection.

[Entropee]

Well so far we think that the Universe is finite, but with no boundaries. I really hate the word infinity >.< It's so unbelievably unfathomable, and people just toss it around as if it's just a large number. Good thoughts on the subject though.

[Chalnoth]

Well so far we think that the Universe is finite, but with no boundaries.
I don't know who this "we" is, but so far as I am aware there is no consensus on this. Currently there just is insufficient evidence to say anything more than, "the universe is very, very big."

[JnWaco]

So there might not be another Earth where I'm dating Jennifer Aniston? Dang.

[Chalnoth]

So there might not be another Earth where I'm dating Jennifer Aniston? Dang.
Well, just because you can imagine it doesn't mean it's possible. Consider, for a moment, how many opportunities you have had to get to know a beautiful actress. Probably not very many.

If, in this parallel world, you were the sort of person that had a life where you were in at least occasional contact with beautiful actresses, would your life be so different that that person even count as being you in the first place?

[Entropee]

I don't know who this "we" is, but so far as I am aware there is no consensus on this. Currently there just is insufficient evidence to say anything more than, "the universe is very, very big."

Okay okay.. by "we" I mean Stephen Hawking haha

And if our anti particles formed anti people on an anti earth we could have identical twins down to the last particle...maybe... ;P

[Chalnoth]

Okay okay.. by "we" I mean Stephen Hawking haha
I'm pretty sure if you asked him straight up he'd say basically the same thing I just did. He did, of course, present the no boundary proposal (where the universe has no boundary either in time or in space), but that doesn't mean he would go so far as to claim that he knows it's true. He may think it likely (it's his idea, after all), but I doubt he'd go that far.

And if our anti particles formed anti people on an anti earth we could have identical twins down to the last particle...maybe... ;P
Well, we don't have anti-particles. That's one of the requirements of our cosmological observations: that early-on, there was a very small breaking of the symmetry between matter and antimatter.

[Entropee]

Well yes I agree with you thats why i said its what he "thinks" not knows, and would YOU go so far as to say that you "know" we dont have anti-particles? There may not be symmetry between matter and antimatter but It's still somewhere.

[Chalnoth]

Well yes I agree with you thats why i said its what he "thinks" not knows, and would YOU go so far as to say that you "know" we dont have anti-particles? There may not be symmetry between matter and antimatter but It's still somewhere.
Yes, because we've actually looked for them. They're not out there. Basically, if the matter and anti-matter were physically separated, then you'd occasionally get clumps of matter running into clumps of anti-matter, causing rather large explosions. We don't see any of that.

Furthermore there's the problem that around the time of the emission of the CMB, our universe was extremely uniform, so that there was no way that normal matter and anti-matter could have been out of contact with one another.

[Entropee]

Thats true, but why then was the early universe so hot?

Also an unrelated question maybe you can answer for me, if most of the universe is hydrogen, why didnt all the hydrogen undergo nuclear fusion when the universe was as big as a baseball? Was it because the particles were different at the time?

[Chalnoth]

Thats true, but why then was the early universe so hot?
It appears to be a result of the end of inflation. Basically, whatever it was that drove inflation had to have a whole lot of energy. When it decayed, it reheated our universe to a tremendous temperature.

Also an unrelated question maybe you can answer for me, if most of the universe is hydrogen, why didnt all the hydrogen undergo nuclear fusion when the universe was as big as a baseball? Was it because the particles were different at the time?
Right, if the expansion rate would have been slower then, it would have. It would have progressed all the way to producing iron and that'd be the most common element.

But this takes time. First the protons and neutrons condense out of the quark-gluon plasma, so you have hydrogen right away. Then you start to make helium from the hydrogen. Then you start to make heavier and heavier elements.

As it turns out, the expansion rate was such that the universe cooled to where the nuclear fusion basically stopped by the time there was only around 25% helium sitting around, and long before more than trace amounts of anything else formed. Incidentally, this is even faster than it sounds, because much of the helium stemmed from the neutrons that were around early-on (when you have nothing but protons around, fusion takes a heck of a lot of energy, as you have to convert protons to neutrons, and because they repel one another, but when you have lots of neutrons sitting around it's much easier).

[Entropee]

O.o I thought we didnt have protons and neutrons in the early universe, wasn't there lots of particle decay so they would be different now?

[Chalnoth]

O.o I thought we didnt have protons and neutrons in the early universe, wasn't there lots of particle decay so they would be different now?
Oh, well, in the very early universe there weren't any. But when the quark-gluon plasma cooled, well, protons and neutrons were the particles they condensed into: they're the lightest baryons. Heavier baryons are unstable and would have quickly decayed into protons and neutrons.

[Entropee]

About how long did it take for the quark-gluon plasma to cool?

[Chalnoth]

About how long did it take for the quark-gluon plasma to cool?
Well, either way I don't know off the top of my head and you might be able to find it yourself as quickly as I could off of Google, but what specifically do you mean by this?

That is, are you asking how long the process of nucleosynthesis took, from the time the protons/neutrons condensed out of the plasma to the time that fusion stopped? Or are you asking how long after the end of inflation that this occurred?

[Entropee]

How long after the end of inflation is more what i meant. But yeah i could google it lol.

[Chalnoth]

How long after the end of inflation is more what i meant. But yeah i could google it lol.
Hehe :) Yeah, I actually looked it up to. It's about three minutes ;)

[Entropee]

Wow nice haha, did you see my post on your profile?

[JnWaco]

Well, just because you can imagine it doesn't mean it's possible. Consider, for a moment, how many opportunities you have had to get to know a beautiful actress. Probably not very many.

If, in this parallel world, you were the sort of person that had a life where you were in at least occasional contact with beautiful actresses, would your life be so different that that person even count as being you in the first place?

lol, you sure know how to pour water on a nice thought!!!!!

just kidding

[Chalnoth]

lol, you sure know how to pour water on a nice thought!!!!!

just kidding
Haha, well, I think that it still can be extremely interesting.

Imagine, for a moment, just walking down the street. If we just take the part of your wave function that you know about today, and imagine all of the future parts (using the many worlds interpretation, of course), then those future parts will likely follow nearly the same but slightly different trajectories, for a while. I imagine it like a blurring of myself, some parts slightly ahead, some slightly behind. Some slightly to the left, some slightly to the right, etc. So the different parts of my wave function are slowly, very slowly dispersing. It might take many trips outside the house before anything interesting happens.

But then something interesting does happen: a car, going too fast, almost hits me. Well, it almost hits the "me" that I see, but there are other me's that were in slightly different places: some of them were just a little bit too close to the car, and got smacked. Suddenly, what were once very similar worlds become very different.

In another situation, something very similar is happening, but the event that causes the divergence is, say, I'm not paying attention to where I'm going, and I almost run into a pretty girl. I manage to apologize for the incident, strike up a conversation, and we start dating. Of course, that's just the "me" that I observe: some of the me's in other worlds either are far enough away that they don't almost run into her, or are close enough that they actually run into her and just end up pissing her off. In some others, the conversation takes a slightly different turn and we never see each other again. Etc. etc.

So there might well be people out there who, when I was a child, were still part of my wave function, the part that I remember, but who today have extremely different lives.

[Axuality]

If the universe is infinite, does that mean that everything exists somewhere, besides obviously impossible things like a star that contains oxygen but doesn't contain oxygen or a 4-sided triangle?


You must define the word 'infinite' before you can validly ask this question, and before anyone can validly answer it.

Does that make sense?

And if you're able to define it to your satisfaction, then you'll have answered your own question, I believe. :)

[Chalnoth]

You must define the word 'infinite' before you can validly ask this question, and before anyone can validly answer it.
That's easy: if the universe is infinite in space, then even if one could travel much faster than the speed of light, one could travel forever without ever crossing one's path.

Another way of saying it is that if you could write down coordinates for the entire universe, then you'd never reach a number in those coordinates that was "beyond" the universe.

[JnWaco]

Haha, well, I think that it still can be extremely interesting.

Imagine, for a moment, just walking down the street. If we just take the part of your wave function that you know about today, and imagine all of the future parts (using the many worlds interpretation, of course), then those future parts will likely follow nearly the same but slightly different trajectories, for a while. I imagine it like a blurring of myself, some parts slightly ahead, some slightly behind. Some slightly to the left, some slightly to the right, etc. So the different parts of my wave function are slowly, very slowly dispersing. It might take many trips outside the house before anything interesting happens.

But then something interesting does happen: a car, going too fast, almost hits me. Well, it almost hits the "me" that I see, but there are other me's that were in slightly different places: some of them were just a little bit too close to the car, and got smacked. Suddenly, what were once very similar worlds become very different.

In another situation, something very similar is happening, but the event that causes the divergence is, say, I'm not paying attention to where I'm going, and I almost run into a pretty girl. I manage to apologize for the incident, strike up a conversation, and we start dating. Of course, that's just the "me" that I observe: some of the me's in other worlds either are far enough away that they don't almost run into her, or are close enough that they actually run into her and just end up pissing her off. In some others, the conversation takes a slightly different turn and we never see each other again. Etc. etc.

So there might well be people out there who, when I was a child, were still part of my wave function, the part that I remember, but who today have extremely different lives.


I'm a layman, just getting into physics, but that makes sense. So are there infinite universes, or just one universe that is infinite. I saw a special on the Discovery channel where they said there were an infinite number of universes in the "multiverse".

How was that proven (or derived)?

[Axuality]

That's easy: if the universe is infinite in space, then even if one could travel much faster than the speed of light, one could travel forever without ever crossing one's path.

Another way of saying it is that if you could write down coordinates for the entire universe, then you'd never reach a number in those coordinates that was "beyond" the universe.

You're a smart guy/girl obviously, and thanks for your post.

But he didn't SAY 'infinite in space', like you did. He said just 'infinite'.

But despite that, no matter how fast you travel, you would never cross your own path in a FINITE universe, if the universe was constantly creating itself faster than you could get there.

And as to your second definition, you can't write down the coordinates for an entire infinite universe. And even if you tried, you might never reach a number outside the universe because the universe may be growing faster than you can chart it...- while still remaining currently finite.

So again, like the original poster, you can't really use the word infinite even in a phrase such as 'infinite in space', until you define the word infinite. As soon as you do, then you'll realize that infinite has but one definitive meaning. All others are derived for context.

Not to mention after all, that you gave descriptions of 'infinite in space', not a definition of 'infinite in space'.

So my proposal remains- You must define 'infinite' before you use the term.

[Axuality]

If the universe is infinite, does that mean that everything exists somewhere, besides obviously impossible things like a star that contains oxygen but doesn't contain oxygen or a 4-sided triangle?

Yes Richard87, if the universe is TRULY infinite, it means that it contains all things real.

And if you're wondering if you're dating Jennifer Aniston somewhere, I can tell you 'yes, you are', and tell you how to find that "part" of the universe, but don't ask please. :)

[Chalnoth]

You're a smart guy/girl obviously, and thanks for your post.
Guy, actually :)

But he didn't SAY 'infinite in space', like you did. He said just 'infinite'.

But despite that, no matter how fast you travel, you would never cross your own path in a FINITE universe, if the universe was constantly creating itself faster than you could get there.
True, but that's usually what people mean. And that's why I said, "even if you could move at arbitrary speed" :) Perhaps a better way of stating it is that if you could freeze the expansion, then you could travel forever without crossing your own path in an infinite universe.

And as to your second definition, you can't write down the coordinates for an entire infinite universe. And even if you tried, you might never reach a number outside the universe because the universe may be growing faster than you can chart it...- while still remaining currently finite.
Well, in general this is true, you can't. But in special cases it's entirely possible (because the real numbers are also infinite). A flat, uniformly-expanding universe would be one example, where simple co-moving coordinates work just fine. The infinite extent actually doesn't affect this.

Mathematically, infinity is quite well-defined. See the extended real number line here (http://en.wikipedia.org/wiki/Extended_real_number_line).

[JnWaco]

I was reading about infinity - and aren't there differing orders of infinity, and even sets of infinite numbers that still exclude other numbers?

Like the set of all even numbers is infinite. But it does not include the number 1, 3, 5, 7, etc. So even if the universe was infinite, there could still be an "everthing" that doesn't exist?

Perhaps this is more of a philosophical question.

[Chalnoth]

I was reading about infinity - and aren't there differing orders of infinity, and even sets of infinite numbers that still exclude other numbers?

Like the set of all even numbers is infinite. But it does not include the number 1, 3, 5, 7, etc. So even if the universe was infinite, there could still be an "everthing" that doesn't exist?

Perhaps this is more of a philosophical question.
Well, there could be an infinite number of possible things that still don't exist. However, I don't think that would count as "everything".

[JnWaco]

Well, there could be an infinite number of possible things that still don't exist. However, I don't think that would count as "everything".

Well, I was referring to the original poster's question - poor choice of words on my part- I just meant that an infinite universe could still not have everything exist, and in fact there could be an infinite number of things that wouldn't exist.

[SpaceTiger]

However, there may be other reasons to believe that all possibilities are realized, mainly stemming from quantum mechanics, where we find, for instance, that if there is the possibility of matter inhabiting a region of space, then particles of that sort of matter will necessarily pop in and out of the vacuum. Another way of saying this is that in quantum mechanics, there mere possibility of existence forces existence. So it is not unreasonable to suspect that perhaps all possibilities must actually be realized.

This has been my increasing feeling as well; that is, that there is no actual difference between possibility and actuality. It even negates the necessity for "meaning" or an "origin" -- things are simply because they can be. To my knowledge, however, it's still not the mainstream interpretation of quantum mechanics in the physics community (still Copenhagen?), though it's not clear how useful a "mainstream" stance on philosophy is.

Anyway, so that I don't venture too far off-topic, to address the original question... we still can't say whether the universe is infinite or finite and it's possible we'll never know. If the universe is finite, it's likely that our particle horizon back to inflation (the largest comoving distance we can possibly observe) is a good deal smaller than the full extent of the universe. If inflationary theory is correct, the rapid expansion of the scale factor in the early universe would have caused our effective horizon to contract from its pre-inflation size and render much of the universe unobservable.

[Chalnoth]

This has been my increasing feeling as well; that is, that there is no actual difference between possibility and actuality. It even negates the necessity for "meaning" or an "origin" -- things are simply because they can be. To my knowledge, however, it's still not the mainstream interpretation of quantum mechanics in the physics community (still Copenhagen?), though it's not clear how useful a "mainstream" stance on philosophy is.
So far as I am aware, among people that have actually thought about this in depth, the many-worlds interpretation is predominant. I think the Copenhagen interpretation is just a result of the "shut up and calculate" school of thought, where a large number of physicists just don't want to bother with these sorts of details, and would rather just get to work learning other things.

More recently, however, quantum computing research has forced many physicists to look in more detail at the specific nature of collapse, and so I expect that the "shut up and calculate" school won't last very long, as the Copenhagen interpretation doesn't actually say what happens at the boundary of collapse. It doesn't say when collapse happens, or how it happens.

There are also a few other schools of thought, but as far as I know they are strong minority views.

Anyway, if you're interested in reading a bit more on the "anything that can happen does happen" possibility, you may be interested in this paper by Max Tegmark:
http://arxiv.org/pdf/0704.0646

It's highly speculative, and would be extremely difficult to find any evidence in favor of it, but I find it quite the intriguing idea.

[SpaceTiger]

Anyway, if you're interested in reading a bit more on the "anything that can happen does happen" possibility, you may be interested in this paper by Max Tegmark:
http://arxiv.org/pdf/0704.0646

Thanks for the link. I knew that Max was vocal on these issues, but hadn't seen his paper.

[oldman]

If the universe is infinite, does that mean that everything exists somewhere

No, of course not.

Some infinities are bigger than others, and the infinity of distinguishable configurations of things is a factorial kind of infinity that is always much, much bigger than the infinity of the number of things.

Think of building a universe as a collection of things, starting with a just a few. As your universe grows the number of ways the things can be arranged differently grows very much faster than their number. So all possible configurations (everything) is something that can never be realised. You needn't even struggle with the impossibility of imagining infinity. And that old idea of monkeys typing Hamlet, given enough time, is nonsense for much the same kind of reason.

[Chalnoth]

And that old idea of monkeys typing Hamlet, given enough time, is nonsense for much the same kind of reason.
Not really, because that idea is based upon simple probability, and assumes that the monkeys are typing in a purely random fashion. If that's the case, then as time gets large enough, the probability that they successfully type hamlet will approach one. Of course, the time required to do that is unbelievably large.

[Freeman Dyson]

In an infinite universe, anything above zero probability happens an infinite amount of times. If the universe were infinite it would take an infinite amount of time to figure that out. The finite hides the infinite.

[Chronos]

About 3 minutes after the big bang the universe was cool enough for elementary particles [e.g., protons] to begin to form.

[ViewsofMars]

I'm going to play around with what you said Freeman. :biggrin: (I'm in one of those kind of moods today. I hope you don't mind. )

In an infinite universe [with dark energy], anything above zero probability happens an infinite amount of times [due to dark energy].

I've inserted within your thoughts my thoughts noted within the brackets.

If the universe were infinite it would take an infinite amount of time to figure that out. The finite hides the infinite.

Interesting. Two comments:

1. "However, the results of the WMAP mission and observations of distant supernova have suggested that the expansion of the universe is actually accelerating which implies the existence of a form of matter with a strong negative pressure, such as the cosmological constant. This strange form of matter is also sometimes referred to as the "dark energy". If dark energy in fact plays a significant role in the evolution of the universe, then in all likelihood the universe will continue to expand forever." (From NASA, Is the Universe Infinite?
http://map.gsfc.nasa.gov/universe/uni_shape.html)

2. "Since light travels at a finite speed, distant objects are seen as they existed in the past. We see the Sun not as it is now, but how it was eight minutes ago. (The Sun is eight light minutes away from the Earth). We see the nearby stars as they were several years ago. We see Andromeda, the nearest spiral galaxy as it was roughly 2.5 million years ago. Thus, the most distant objects that we see are the oldest objects that we can directly detect.

" Quasars are the most distant distinct objects that astronomers have been able to detect. In a region smaller than our solar system, a quasar emits more light than our entire Milky Way galaxy. Quasars are believed to be supermassive black holes, whose masses exceed that of a million Suns, and whose pull is swallowing gas and stars from their host galaxies. They shine brightly by converting the gravitational energy of the infalling material into light. The most distant quasars are seen at a time when the universe was one tenth its present age, roughly a billion years after the Big Bang." (From NASA- When Did the First Cosmic Structures Form?
http://map.gsfc.nasa.gov/universe/rel_firstobjs.html)

[Dmitry67]

No, of course not.

Some infinities are bigger than others, and the infinity of distinguishable configurations of things is a factorial kind of infinity that is always much, much bigger than the infinity of the number of things.

Think of building a universe as a collection of things, starting with a just a few. As your universe grows the number of ways the things can be arranged differently grows very much faster than their number. So all possible configurations (everything) is something that can never be realised. You needn't even struggle with the impossibility of imagining infinity. And that old idea of monkeys typing Hamlet, given enough time, is nonsense for much the same kind of reason.

Max Tegmark does not agree with you:

http://space.mit.edu/home/tegmark/multiverse.html

How many parallel universes are there?
Why must we have duplicates?

From Richard Reeves, valueprint@earthlink.net, April 18, 2003 14:23:31
Q: Given infinity, why isn't it equally plausible that the worlds within it would express infinite variety, rather than repetition
The answer is that there are only a finite number of possible states that a Hubble volume can have, according to quantum theory. Even classically, there are clearly only a finite number of perceptibly different ways it can be.

How rigorous is this?

From Bert Rackett, bertrckt@pacbell.net, Sat Apr 19 22:22:13 2003
Q: I very much enjoyed reading your Scientific American and Science and Uitimate Reality papers, but I am entirely befuddled about your estimates for likely distance of an identical environment. You claim that the volume may be completely defined by a (very long) list of binary values denoting the presence or absence of a proton, but this of course oversimplifies things.
A: Although classical physics allows an infinite number of possible states that a Hubble volume can be in, it's a profound and important fact that quantum physics allows only a finite number. The numbers I mentioned in the article, like 10^10^118 meters, were computed using the exact quantum-mecanical calculation, and the classical stuff about counting protons in a discrete lattice arrangement was merely thrown in as a pedagogical example to give a feel where the numbers come from, since that turns out to give the same answer.

Why must all regions have duplicates, not just one?
From Jeffery Winkler, jeffery_winkler@mail.com, Oct 13, 2003, at 0:58
Q: Just because something is infinite, does not mean that all possibilities are realized. The number pi is infinitely long, pi = 3.14159... and in that case, all combinations of digits are realized. However, the number 1/3, converted into a fraction, is also infinitely long, 1/3 = .33333... and in that case, all combinations of digits are not realized.
A: That's correct: infinite space alone guarantees only that SOME Hubble volume will have a duplicate, not that our own will. However, if (as in the current cosmological standard model) the cosmic density fluctuations originate from quantum fluctuations during inflation, their statistical properties DO guarantee that our (and indeed every) Hubble volume has a duplicate.
Is there a countable or uncountable infinity of universes?

Is it countable even with continuous wave functions?
From David Fotland, fotland@smart-games.com, August 3, 2003 21:09:49
You argued that the total number of possible states in a universe is finite, so if the total of all universes is infinite, then every possible universe must exist. I understand that quantum states have discrete vales, but the wave function is a continuous function. Can't the probabilities that give the possible locations of particles have any real value?
Interestingly, they can't: you can prove that in a finite volume, there's only a discrete number of allowed quantum wavefunctions. If the energy is finite, it's even a finite number.
But even a hydrogen atom has infinitely many states!
From Attila Csoto, csoto@matrix.elte.hu, Wed Mar 17 12:59:29 2004
Q: You say in your papers that the number of possible quantum states within the Hubble-volume is finite. I understand your argument, but there is a problem which puzzles me. If we single out one hydrogen atom in our Hubble volume, it has itself an infinite number of different bound states. So one could imagine a Hubble sphere next to ours which is the same as ours except that this hydrogen atom iis not in its ground state but in the next excited state, and in the next sphere in the next higher state, etc. These universes differ from each other by a tiny amount of energy but I don't think that this should matter. So, my question is: how can we have a finite number of possible quantum states in our sphere, if one hydrogen atom already has an infinite number of possible bound states?
A: There's infinitely many bound states if only space is truly infinite. There's in fact a beautiful old paper by Erwin Schrödinger deriving the exact solutions for a hydrogen atom in a closed finite Universe, showing that in this case, the number of bound states is finite.

[ViewsofMars]

Dmitry67, you have presented Max Tegmark whose last publication was from 2003, Richard Reeves (2003), Bert Rackett (2003), Jeffery Winkler (2003), David Fotland (2003), Attila Csoto (2004).

The year is 2009.:biggrin: We have come a long way since 2003-2004. You may wish to review the topic “offshoot from 'Theoretically how far can one see in the universe'” p.g. 3, and look at my contributions (msg’s 40 and 42) that have the most current evidence (2008 and 2009) that continues to support the Big Bang Theory, which is the standard model. I've provided the link below for you and others to explore.

I should also mention that Steven Weinberg on July 7, 2009 gave a lecture at CERN. He talked about inflation. Near the end of his presention he did say, "The world is really what we've always known, the standard model plus relativity." He wasn't trying to discourage string theorists yet he knows as a scientist that observations are essential if you are going to call it SCIENCE. :smile:

http://www.physicsforums.com/showthread.php?t=338032&page=3

[Dmitry67]

It appears that you are not familiar with the most important article: http://arxiv.org/abs/0704.0646 - really genious!
and year is 2007 - not 2003. Do you have any other arguments against his logic except the year of publication?

[ViewsofMars]

It appears that you are not familiar with the most important article: http://arxiv.org/abs/0704.0646 - really genious! and year is 2007 - not 2003.
Don't be curt or snippy with me. Dmitry67, I was going off the first link you gave when mentioning Max Tegmark. I didn't see this abstract you are now presenting on his website. The *abstract* by Max Tegmark is entitled, "The Mathematical Universe". As we know, math isn't science. :smile: Futhermore, it is a hypothesis of his.

Do you have any other arguments against his logic except the year of publication?

The year of publication is important as you will note with WAMP. I think it is important to see the current information. I've earlier provided you a link to another topic for you to review.

I know now that Max Tegmark's has a *hypothesis* that has been submitted to Cornell University. I'm sorry to disappoint you, but I'm not a fan of his "consciousness" and "parallel universes". I don't find his hypothesis in NASA. Also, hypothesis don't make it into internationally known peer-reviewed journals such as Science and Nature.:smile:

[Dmitry67]

At first, could you clarify what do you mean by "math isn't science"? Do you mean that we cant prove self-consistency of any axiomatic system, or something else?

[Chalnoth]

Dmitry67,

Yes, I've heard this argument about "copy universes" before. I don't think it necessarily holds.

Here is the problem, as I see it: quantum mechanics doesn't guarantee that absolutely everything happens. It just says that many things happen. For instance, let's say I take a simple harmonic oscillator, and prepare it in the following state:

\mid \psi \rangle = \frac{1}{\sqrt{2}} \mid E_1 \rangle + \frac{1}{\sqrt{2}} \mid E_2 \rangle

Here we have a system in a mixture of two states. If I take a measurement of the energy at t=0, I will obtain with 50% probability E_1, and E_2 with 50% probability.

Now step back and consider what I might be able to conclude if I didn't know how the state was prepared, but only am aware of my measurement of the state. For instance, if I measure E_2, what can I conclude about the original state? Can I conclude that E_3 was also represented? That some other "me" observed E_3 while I observed E_2? Certainly not! I do know that whatever state the system was in, it was in a specific state, and my measurement of E_2, while not ruling out the possibility of a component of the wavefunction inhabiting E_3, there is no reason to believe that E_3 was represented (or E_1, for that matter).

Therefore I claim that even with taking quantum fluctuations into account, I don't see why all possibilities need to be represented. Regions of the universe are still due to unitary evolution of a quantum mechanical wave function. They are still deterministically dependent upon the initial conditions. And so even though many things surely do happen, we can't say with confidence that all possible things do.

However, that said, I do think that it is entirely possible that all possible things do happen. I particularly like Tegmark's "mathiverse" idea, for instance. I just don't think we can conclude from what we know today that it's actually true that all things possible do happen.

[rasp]

If the universe is infinite, does that mean that everything exists somewhere, besides obviously impossible things like a star that contains oxygen but doesn't contain oxygen or a 4-sided triangle?

The answer to your question depends on what you mean by everything. If by everything you mean all that is, then everything fits into a finite Universe, an infinite Universe is not needed. If your definition of everything is what is existent and also on what doesn't exist but could exist, then an infinite universe doesn't help you, you need either a many world interpretation, or a Hilbert Space in QM.

[xantox]

Therefore I claim that even with taking quantum fluctuations into account, I don't see why all possibilities need to be represented.
Do you mean, by questioning the assumptions of ergodicity and equilibrium of the inflationary patches?

[Chalnoth]

Do you mean, by questioning the assumptions of ergodicity and equilibrium of the inflationary patches?
I'm pretty sure those are taken to be approximating assumptions.

[xantox]

I'm pretty sure those are taken to be approximating assumptions.
Yes, they are just assumptions – just making sure whether you agree that if they are true then "copy universes" would hold too, or if your argument was another one. Although those assumptions are supported by unitarity, quantum gravity effects such as backreaction on the metric could indeed subtly change the picture.

[Entropee]

I'm not sure anything in our universe at all can lead us to believe anything at all about a "different" universe. Just my opinion.

[Chalnoth]

I'm not sure anything in our universe at all can lead us to believe anything at all about a "different" universe. Just my opinion.
As long as we believe that the laws of physics are invariant of where you happen to be, yes, we can say some things about regions which are outside of our observable bubble.

[ViewsofMars]

Replying to Dmitry67, we have Math teachers and Science teachers. The two are distinctly different. Science is based on the scientific method, whereas Mathematics is not.

[Entropee]

As long as we believe that the laws of physics are invariant of where you happen to be, yes, we can say some things about regions which are outside of our observable bubble.

Yes inside our "observable bubble". I just meant that we have to reason to believe our universes laws of physics hold true in a different universe (whatever that means).

[Freeman Dyson]

"'the universe must go through a calculable number of combinations in the great game of chance which constitutes its existence . . . In infinity, at some moment or other, every possible combination must once have been realized; not only this, but it must also have been realized an infinite number of times."

-nietzsche

[Chalnoth]

Yes inside our "observable bubble". I just meant that we have to reason to believe our universes laws of physics hold true in a different universe (whatever that means).
Typically it just means something outside of our observable bubble. And while we may have reason to believe that there are different effective low-energy laws of physics, there are good reasons to believe that the fundamental laws remain the same no matter what.

[sokrates]

What are those good reasons to believe the laws of physics are the same outside our bubble?

I have a feeling whatever you'll argue is going to be exclusively about "observable" bubble, almost by def.

[Chalnoth]

What are those good reasons to believe the laws of physics are the same outside our bubble?
One way to look at it is that if there is stuff outside our observable universe (which there almost certainly is), then it was at one point connected to our observable universe. If it didn't follow the same fundamental laws when it was in contact, then you'd have a contradiction.

[Dmitry67]

I know now that Max Tegmark's has a *hypothesis* that has been submitted to Cornell University. I'm sorry to disappoint you, but I'm not a fan of his "consciousness" and "parallel universes". I don't find his hypothesis in NASA. Also, hypothesis don't make it into internationally known peer-reviewed journals such as Science and Nature.:smile:

Of course there is no proof for his hypotesis. But compare it to the Smolins evolving law: while Max Tegmarks works is quite strict and logical (and in his article he gave answers to most of the questions I found here, so it was enough just to quote his original work) Smolins evolving law is a pure handwaving.

Also, Max Tegmarks hypotesis has several falsifiable predictions so it can be testes in a future. I think this is the best we have for now.

[Dmitry67]

Dmitry67,
Here is the problem, as I see it: quantum mechanics doesn't guarantee that absolutely everything happens.

Yes, definitely, it is interpretation-dependent. In a local region "everything happens" only in MWI.

But interestingly enough, an argument about our distant 'copies' does not depend on MWI and even more, even if you insist that some possible configurations are never realized then there are even MORE copies!

Because if you fill the infinite void with all possible configurations, you will soon ran out of distinct ones (check Max Tegmarks Q&A I posted before). If you insist that only a subset of possible configurations is used, then you will run out of configurations even sooner!

[Dmitry67]

Replying to Dmitry67, we have Math teachers and Science teachers. The two are distinctly different. Science is based on the scientific method, whereas Mathematics is not.

You had probably bad teachers.
You agruments are strange: year of publication, your personal bad luck with teachers.

Can you point an exact place in max Tegmark's logic (in Q&A) which is wrong, as you believe? And explain, why do you think so?

[Dmitry67]

One way to look at it is that if there is stuff outside our observable universe (which there almost certainly is), then it was at one point connected to our observable universe. If it didn't follow the same fundamental laws when it was in contact, then you'd have a contradiction.

Yes.
It is more tricky in an accelerating expanding universe with consmological horizons. In such universe some places will NEVER be in causal contact with each other.

However, still you can define a sequence of intersecting bubbles B0...Bn, so if laws are different in B0 and Bn, there must be a bubble Bx (0<=x<=n) where both laws are effective at the same time.

The same argument applies not only in space but it time, law can not 'change' in time, for that reason I believe that Smolin's evolving law is a nonsense.

[Chalnoth]

Yes, definitely, it is interpretation-dependent. In a local region "everything happens" only in MWI.
No, I'm speaking purely in terms of MWI here. Even in that case, not everything necessarily happens. MWI just takes the unitarity of the wavefunction seriously: it evolves forward in time according to the equations of motion, with no collapse. This doesn't indicate that everything happens, just that many different things do.

I don't think you can take currently-known quantum mechanics and conclude that everything happens: you still have to add it in as an additional assumption.

Because if you fill the infinite void with all possible configurations, you will soon ran out of distinct ones (check Max Tegmarks Q&A I posted before). If you insist that only a subset of possible configurations is used, then you will run out of configurations even sooner!
Ah, yes, well, if the universe truly is infinite in extent, then obviously there will be an infinite number of copies. However, we don't know that the universe is infinite in extent.

[Chalnoth]

Yes.
It is more tricky in an accelerating expanding universe with consmological horizons. In such universe some places will NEVER be in causal contact with each other.
In the future. But in the past they would have been in contact (this would have been during the inflationary epoch for the most distantly-separated components of our universe).

The same argument applies not only in space but it time, law can not 'change' in time, for that reason I believe that Smolin's evolving law is a nonsense.
Well, I'm pretty sure that all serious considerations of evolving or otherwise changing physical laws are actually just talking about the effective low-energy physics. They tend to still rely upon an underlying fundamental theory that is quite invariant. But this isn't really saying something profound about our universe: Andy Albrecht and Alberto Iglesias showed a couple of years back that if you take a random, time-varying Hamiltonian, and simply invoke the clock ambiguity, you can always find a trajectory in this space that leaves the Hamiltonian time-invariant. That is to say, just the fact that we can change coordinates means that it's always possible to write down time-invariant laws of physics.

Here's one of their relevant papers:
http://arxiv.org/abs/0805.4452

[Dmitry67]

1
No, I'm speaking purely in terms of MWI here. Even in that case, not everything necessarily happens. MWI just takes the unitarity of the wavefunction seriously: it evolves forward in time according to the equations of motion, with no collapse. This doesn't indicate that everything happens, just that many different things do.

I don't think you can take currently-known quantum mechanics and conclude that everything happens: you still have to add it in as an additional assumption.

2
Ah, yes, well, if the universe truly is infinite in extent, then obviously there will be an infinite number of copies. However, we don't know that the universe is infinite in extent.

1 Agreed if everything = everything which does not violate any laws
So, everything does not mean that e can dacay, because it would violate the conservation of charge. However, if we monitor a single neutron then it can decay on the 1st second, 2nd, 3rd, ... So MWI insist that there MUST be copies observing a decay on any Nth second. If you say, "decay was possible on 55th second but that branch is actually missing" then you are adding something new, some 'branch scissors' and Ocamm is against you

2 What are the latest observational results?

[Chalnoth]

1 Agreed if everything = everything which does not violate any laws
So, everything does not mean that e can dacay, because it would violate the conservation of charge. However, if we monitor a single neutron then it can decay on the 1st second, 2nd, 3rd, ... So MWI insist that there MUST be copies observing a decay on any Nth second. If you say, "decay was possible on 55th second but that branch is actually missing" then you are adding something new, some 'branch scissors' and Ocamm is against you
Well, right, I'm not saying that. But I think my post #45 makes it clear what I'm talking about. Basically, whatever the latter configuration of the wavefunction is depends upon the former configuration, but as we only have information about our own component of the wavefunction, and don't necessarily have information about the former configuration, we can't necessarily say which other things happen.

2 What are the latest observational results?
Well, basically it comes down to the observed flatness and homogeneity of our universe. The observed homogeneity means that the universe extends significantly beyond what we can see (if it stopped, we should see some effect of that). This is brought down to something more objective with the average curvature, as with that we can make an approximate lower bound on the overall size of our universe. If we make the assumption that our universe is a sphere, for example, then measuring the curvature gives us limits on the size of that sphere. If it isn't a sphere, then it's likely much larger (though not necessarily). So we can get at least a rough lower-limit on the size by constraining the curvature, and so far we've constrained it to within about 1% from flat. From this we can get a very rough lower bound on the size at somewhere in the range of two orders of magnitude larger than our observable region.

[ViewsofMars]

Backing up Chalnoth's observations.

The Wilkinson Microwave Anisotropy Probe (WMAP) is a NASA Explorer mission that launched June 2001 to make fundamental measurements of cosmology -- the study of the properties of our universe as a whole. WMAP has been stunningly successful, producing our new Standard Model of Cosmology. WMAP continues to collect high quality scientific data.

WMAP's Top Ten

1. NASA's Wilkinson Microwave Anisotropy Probe (WMAP) has mapped the Cosmic Microwave Background (CMB) radiation (the oldest light in the universe) and produced the first fine-resolution (0.2 degree) full-sky map of the microwave sky

2. WMAP definitively determined the age of the universe to be 13.73 billion years old to within 1% (0.12 billion years) -as recognized in the Guinness Book of World Records!

3. WMAP nailed down the curvature of space to within 1% of "flat" Euclidean, improving on the precision of previous award-winning measurements by over an order of magnitude

4. The CMB became the "premier baryometer" of the universe with WMAP's precision determination that ordinary atoms (also called baryons) make up only 4.6% of the universe (to within 0.1%)

5. WMAP's complete census of the universe finds that dark matter (not made up of atoms) make up 23.3% (to within 1.3%)

6. WMAP's accuracy and precision determined that dark energy makes up 72.1% of the universe (to within 1.5%), causing the expansion rate of the universe to speed up. - "Lingering doubts about the existence of dark energy and the composition of the universe dissolved when the WMAP satellite took the most detailed picture ever of the cosmic microwave background (CMB)." - Science Magazine 2003, "Breakthrough of the Year" article

7. WMAP has mapped the polarization of the microwave radiation over the full sky and discovered that the universe was reionized earlier than previously believed. - "WMAP scores on large-scale structure. By measuring the polarization in the CMB it is possible to look at the amplitude of the fluctuations of density in the universe that produced the first galaxies. That is a real breakthrough in our understanding of the origin of structure." - ScienceWatch: "What's Hot in Physics", Simon Mitton, Mar./Apr. 2008

8. WMAP has started to sort through the possibilities of what transpired in the first trillionth of a trillionth of a second, ruling out well-known textbook models for the first time

9. The statistical properties of the CMB fluctuations measured by WMAP appear "random"; however, there are several hints of possible deviations from simple randomness that are still being assessed. Significant deviations would be a very important signature of new physics in the early universe

10. Since 2000, the three most highly cited papers in all of physics and astronomy are WMAP scientific papers.
NASA Official: Dr. Gary F. Hinshaw
Page Updated: Tuesday, 04-07-2009
http://map.gsfc.nasa.gov/

I've been taught by the most brilliant scientists! :biggrin: I'm done with this topic.

[rasp]

At first, could you clarify what do you mean by "math isn't science"? Do you mean that we cant prove self-consistency of any axiomatic system, or something else?

Can I jump in and say that it is my understanding that infinity is a mathematical concept which doesn't exist in the real world of science, but which may possibly exist (according to mathematical theories).

[Entropee]

One way to look at it is that if there is stuff outside our observable universe (which there almost certainly is), then it was at one point connected to our observable universe. If it didn't follow the same fundamental laws when it was in contact, then you'd have a contradiction.

I think maybe I didn't explain myself very well. I don't mean the laws might be different for things outside our observable universe, I mean outside our ENTIRE universe. Like I said before, "(whatever that means)". There may not be anything outside our universe, just an infinite number of dimensions we cant perceive.

[Chalnoth]

I think maybe I didn't explain myself very well. I don't mean the laws might be different for things outside our observable universe, I mean outside our ENTIRE universe. Like I said before, "(whatever that means)". There may not be anything outside our universe, just an infinite number of dimensions we cant perceive.
It's just a matter of describing things in the right way, then. Even if things vary, it's always going to be possible to describe them based upon some rules that do not. A good example here would be Tegmark's mathiverse: different universes based upon different mathematical structures are unified by the rule that all mathematical structures exist.

[Entropee]

Im gonna look that up that sounds really interesting.

[Chalnoth]

Im gonna look that up that sounds really interesting.
Here is his webpage on the subject, if you're interested:
http://space.mit.edu/home/tegmark/toe_frames.html

Includes links to the more in-depth treatments of this idea.

[Freeman Dyson]

Can I jump in and say that it is my understanding that infinity is a mathematical concept which doesn't exist in the real world of science, but which may possibly exist (according to mathematical theories).

Right. Infinities exist in math. This was debated throughout history for a while but now math is considered to have infinities. I recently read a good book on infinity.

The Infinite Book: A Short Guide to the Boundless, Timeless and Endless

http://www.amazon.com/Infinite-Book-Boundless-Timeless-Endless/dp/0375422277

It covers nearly everything discussed in this thread. From my understanding, when infinities pop up in the physical world, scientists tend to think of them as a flaw in the theory/measurement. Like how the Big Bang shows infinite properties, it is thought that maybe when a proper theory of quantum gravity is applied to the Big Bang, the infinities will be smoothed down to the finite. Scientists generally don't like infinities in the physical world from what this book says. Infinities don't really exist, they are markers of error in our methods.

[ViewsofMars]

It looks like I need to return to this topic. :rofl: This is common knowledge known by reputable scientists. The Big Bang Theory (the new standard model as mentioned in my previous post) often noted as "Cosmic Evolution" (Professor Chaisson (scientist), Wright Center for Science Education - Cosmic Evolution - http://www.tufts.edu/as/wright_center/cosmic_evolution/docs/splash.html) is based on Science! Thank goodness for that! :)

I'm going to add onto my previous posting to this topic, since it now appears to me that more is needed in understanding what is *science*. I don't wish to get into a debate with people here. Hopefully, people will be able to read, understand what they are reading, and learn. :)

I'm providing three quotes from TalkOrigins that apply to our discussion, though I recommend a thorough reading of the article for possible future discussions on various topics in physicsforums.com.


[snip]
What is meant by scientific evidence and scientific proof? In truth, science can never establish 'truth' or 'fact' in the sense that a scientific statement can be made that is formally beyond question. All scientific statements and concepts are open to re-evaluation as new data is acquired and novel technologies emerge. Proof, then, is solely the realm of logic and mathematics (and whiskey). That said, we often hear 'proof' mentioned in a scientific context, and there is a sense in which it denotes "strongly supported by scientific means". Even though one may hear 'proof' used like this, it is a careless and inaccurate handling of the term. Consequently, except in reference to mathematics, this is the last time you will read the terms 'proof' or 'prove' in this article.

[snip]

Now, to answer the question "What is the scientific method?" - very simply (and somewhat naively), the scientific method is a program for research which comprises four main steps. In practice these steps follow more of a logical order than a chronological one:

1.Make observations.
2.Form a testable, unifying hypothesis to explain these observations.
3.Deduce predictions from the hypothesis.
4.Search for confirmations of the predictions;
if the predictions are contradicted by empirical observation, go back to step (2).
Because scientists are constantly making new observations and testing via those observations, the four "steps" are actually practiced concurrently. New observations, even if they were not predicted, should be explicable retrospectively by the hypothesis. New information, especially details of some process previously not understood, can impose new limits on the original hypothesis. Therefore, new information, in combination with an old hypothesis, frequently leads to novel predictions that can be tested further.

Examination of the scientific method reveals that science involves much more than naive empiricism. Research that only involves simple observation, repetition, and measurement is not sufficient to count as science. These three techniques are merely part of the process of making observations (#1 in the steps outlined above). Astrologers, wiccans, alchemists, and shamans all observe, repeat, and measure — but they do not practice science. Clearly, what distinguishes science is the way in which observations are interpreted, tested, and used.

[snip]

In contrast, Newton's scientific theory of universal gravitation makes specific predictions about what should be observed. Newton's theory predicts that the force between two masses should be inversely proportional to the square of the distance between them (otherwise known as the "inverse square law"). In principle, we could take measurements which indicated that the force is actually inversely proportional to the cube of the distance. Such an observation would be inconsistent with the predictions of Newton's universal theory of gravitation, and thus this theory is testable. Many anti-evolutionists, such as the "scientific" creationists, are especially fond of Karl Popper and his falsifiability criterion. These cynics are well known for claiming that evolutionary theory is unscientific because it cannot be falsified. In this article, these accusations are met head on. Each of the evidences given for common descent contains a section providing examples of potential falsifications, i.e. examples of observations that would be highly unlikely if the theory is correct.
[snip]

http://www.talkorigins.org/faqs/comdesc/sciproof.html



The following should be helpful. It is from the United States National Academy of Sciences (Advisors to the Nation on Science, Engineering, and Medicine).

Is Evolution a Theory or a Fact?

It is both. But that answer requires looking more deeply at the meanings of the words "theory" and "fact."

In everyday usage, "theory" often refers to a hunch or a speculation. When people say, "I have a theory about why that happened," they are often drawing a conclusion based on fragmentary or inconclusive evidence.

The formal scientific definition of theory is quite different from the everyday meaning of the word. It refers to a comprehensive explanation of some aspect of nature that is supported by a vast body of evidence.

Many scientific theories are so well-established that no new evidence is likely to alter them substantially. For example, no new evidence will demonstrate that the Earth does not orbit around the sun (heliocentric theory), or that living things are not made of cells (cell theory), that matter is not composed of atoms, or that the surface of the Earth is not divided into solid plates that have moved over geological timescales (the theory of plate tectonics). Like these other foundational scientific theories, the theory of evolution is supported by so many observations and confirming experiments that scientists are confident that the basic components of the theory will not be overturned by new evidence. However, like all scientific theories, the theory of evolution is subject to continuing refinement as new areas of science emerge or as new technologies enable observations and experiments that were not possible previously.

One of the most useful properties of scientific theories is that they can be used to make predictions about natural events or phenomena that have not yet been observed. For example, the theory of gravitation predicted the behavior of objects on the moon and other planets long before the activities of spacecraft and astronauts confirmed them. The evolutionary biologists who discovered Tiktaalik predicted that they would find fossils intermediate between fish and limbed terrestrial animals in sediments that were about 375 million years old. Their discovery confirmed the prediction made on the basis of evolutionary theory. In turn, confirmation of a prediction increases confidence in that theory.

In science, a "fact" typically refers to an observation, measurement, or other form of evidence that can be expected to occur the same way under similar circumstances. However, scientists also use the term "fact" to refer to a scientific explanation that has been tested and confirmed so many times that there is no longer a compelling reason to keep testing it or looking for additional examples. In that respect, the past and continuing occurrence of evolution is a scientific fact. Because the evidence supporting it is so strong, scientists no longer question whether biological evolution has occurred and is continuing to occur. Instead, they investigate the mechanisms of evolution, how rapidly evolution can take place, and related questions.
http://www.nationalacademies.org/evolution/TheoryOrFact.html

We should also be mindful of this from NASA.

Tests of Big Bang: Expansion
NASA Official: Dr. Gary F. Hinshaw (scientists)
Page Updated: Tuesday, 10-14-2008

The Big Bang model was a natural outcome of Einstein's General Relativity as applied to a homogeneous universe. However, in 1917, the idea that the universe was expanding was thought to be absurd. So Einstein invented the cosmological constant as a term in his General Relativity theory that allowed for a static universe. In 1929, Edwin Hubble announced that his observations of galaxies outside our own Milky Way showed that they were systematically moving away from us with a speed that was proportional to their distance from us. The more distant the galaxy, the faster it was receding from us. The universe was expanding after all, just as General Relativity originally predicted! Hubble observed that the light from a given galaxy was shifted further toward the red end of the light spectrum the further that galaxy was from our galaxy.

The Hubble Constant

The specific form of Hubble's expansion law is important: the speed of recession is proportional to distance. The expanding raisin bread model at left illustrates why this is important. [Please view the "expanding raisin bread model" by clinking on the link below.] If every portion of the bread expands by the same amount in a given interval of time, then the raisins would recede from each other with exactly a Hubble type expansion law. In a given time interval, a nearby raisin would move relatively little, but a distant raisin would move relatively farther - and the same behavior would be seen from any raisin in the loaf. In other words, the Hubble law is just what one would expect for a homogeneous expanding universe, as predicted by the Big Bang theory. Moreover no raisin, or galaxy, occupies a special place in this universe - unless you get too close to the edge of the loaf where the analogy breaks down.

The current WMAP results show the Hubble Constant to be 73.5 +/-3.2 (km/sec)/Mpc. If the WMAP data is combined with other cosmological data, the best estimate is 70.8 +/- 1.6 (km/sec)/Mpc.
http://map.gsfc.nasa.gov/universe/bb_tests_exp.html

A review of my mgs. 39 might be helpful. A segment from that post was from a "scientist (physicist) -" from NASA, Is the Universe Infinite? Here is a quote from him, but please review the entire website.

"However, the results of the WMAP mission and observations of distant supernova have suggested that the expansion of the universe is actually accelerating which implies the existence of a form of matter with a strong negative pressure, such as the cosmological constant. This strange form of matter is also sometimes referred to as the "dark energy". If dark energy in fact plays a significant role in the evolution of the universe, then in all likelihood the universe will continue to expand forever." http://map.gsfc.nasa.gov/universe/uni_shape.html

Also, "Mathematicians" are not scientists. "Physicists" are scientists that know mathamatics. :)

Have a good day,
Mars

[Entropee]

Thanks Chalnoth I actually couldnt find it on my own lol

[Axuality]

Can I jump in and say that it is my understanding that infinity is a mathematical concept which doesn't exist in the real world of science, but which may possibly exist (according to mathematical theories).

Of all words, 'infinity' is tied as the most striking example of a word that can have only one definition. For example, I'm sure we all know that the number of integers is not infinite, it is unlimited.

The only definition of infinity which is rational, is 'the summation of all things'. There can be only one infinity,- in much the same way that there can be only one reality, incidentally.

That is important, because upon reflection, it reveals startling things about the nature of reality, which reveals additional startling things about the natures of science and math.

[Chalnoth]

Of all words, 'infinity' is tied as the most striking example of a word that can have only one definition. For example, I'm sure we all know that the number of integers is not infinite, it is unlimited.
Uh, that's not strictly true. Mathematically speaking, different infinities can and often do have rather different characters.

The number of integers, for instance, is called a "countably infinite" number. Any set of numbers which can be mapped one-to-one onto integers is also countably infinite. Sets which cannot be mapped onto the integers (such as the reals) are uncountably infinite, which means that there are, for instance, more real numbers than there are integers (by contrast, there are no fewer positive integers than total integers).

The only definition of infinity which is rational, is 'the summation of all things'. There can be only one infinity,- in much the same way that there can be only one reality, incidentally.

That is important, because upon reflection, it reveals startling things about the nature of reality, which reveals additional startling things about the natures of science and math.
Sorry, but definitions are arbitrary. There is never only one rational definition.

[Axuality]

Uh, that's not strictly true. Mathematically speaking, different infinities can and often do have rather different characters.

The number of integers, for instance, is called a "countably infinite" number. Any set of numbers which can be mapped one-to-one onto integers is also countably infinite. Sets which cannot be mapped onto the integers (such as the reals) are uncountably infinite, which means that there are, for instance, more real numbers than there are integers (by contrast, there are no fewer positive integers than total integers).


Sorry, but definitions are arbitrary. There is never only one rational definition.

Ha ha, you obviously are intelligent, so not for one moment would I forget that.

Perhaps I should have said that there "should" be only one definition of the word 'infinity'.
I know and understand what you told me about 'infinity'. And I recognize that everything you said was correct. What I am saying to you though, is that the understanding of the concept of TOTAL infinity makes impossible the logical use of the word infinity in the phrase "countable infinities". I mean if we want to call a horse a horse, and also call a cow a horse, we can do it. But it makes things less clear, not more clear.

And when you tell me that "definitions are arbitrary" I know what you mean of coures, but I respectfully chuckle to realize that the word 'definitions' is somehow based on the word 'definite' which would make the statement kind of like saying 'definite is arbitrary', which in some sense is rather contradictory. :)

More seriously though, I disagree that there is "never only one rational definition". While on the surface that seems, and IS correct, I'm not on the surface with this definition thing.

In fact, I construe and extrapolate to conclude that IF that statement is true, then by it's own truth, it is not ALWAYS true. Hence it is not true at all.

Forgive me, I don't think I'm 'smarter' than you. I think in fact, that I'm not smart ENOUGH to convey to you that I'm am talking about a slightly different aspect of 'definition' than you are.

I would beg you to simply consider (for just a moment) the definition of 'infinity' as being the entire collection of all things which compose reality (matter, energy, thoughts, et al). --that's what infinity is; what it means. If we want to call a cow a horse, then we can use the word 'infinity' to mean something else also. :)

[Chalnoth]

The problem, Axuality, is that you're abusing what it means to define a word.

First, as I said, there is never any one rational definition: all definitions are arbitrary. And furthermore, words in the English language tend to be extremely context-sensitive. The important point isn't that words have rational definitions, but rather that words are understood. That is to say, words are defined by how they are used by people. This means that if you are to use a word, it is a darned good idea to understand how people will interpret that word.

So when you go and use a definition of infinite as "the summation of all things", that strikes me as rather ridiculous as nobody uses that definition. Infinite is, by large, an intrinsically mathematical term (except when it colloquially used to mean "really really big"). In mathematics, there are a few different classes of infinites. And the fact is, we do not yet know for sure whether or not various parts of our universe match one of these different classes of infinities.

Thus if you want to talk about the "summation of all things", if you wish to be understood, you should use the word "universe" instead of "infinity".

[Axuality]

The problem, Axuality, is that you're abusing what it means to define a word.

First, as I said, there is never any one rational definition: all definitions are arbitrary.


if you want to talk about the "summation of all things", if you wish to be understood, you should use the word "universe" instead of "infinity".

Hi. If we're are going to discuss any further, I need you to understand that I respect your intelligence. Therefore, I will speak as if I know that you will not be offended.

I started a response to you which became too long for you to read and for me to write, so I'm shortening it. :)

I do not agree with what you say.

#1 In the ultimate, the word 'universe' and the word 'infinity' are identical in meaning. If you doubt this, you are not looking large enough.

The universe is larger in scope(not physical scale, but 'scope') than is imagined by physics. Quantum theory is approaching a conclusion on the subject which will substantiate this.

Maybe I should have said in the first place "There is only one infinity". There are many definitions of the word 'infinity', but there is only one infinity. To understand this, you must be able to separate the concept behind a word from the definition of that word. You may well doubt that that is possible or makes sense, but that is okay if you doubt it. ;)

#2 The statement that 'all definitions are arbitrary' is self-contradictory.- much as the statement 'Truth does not exist' is self-contradictory. I'm going to abstain from any attempt at long proof of that, and if you don't choose to believe it, that is okay. :) I had to put it out there. ( let me make a quick offering of "proof" --'if all definitions are arbitrary, or relative, then there ARE no definitions, there are only 'word assignments'. Maybe we need a new definition of the word 'definition'.

Again, you're obviously a smart guy or girl, and I hope I've been able to speak directly and unoffendingly, if not very diplomatically. :)

[Deuterium2H]

I was reading about infinity - and aren't there differing orders of infinity, and even sets of infinite numbers that still exclude other numbers?

Like the set of all even numbers is infinite. But it does not include the number 1, 3, 5, 7, etc. So even if the universe was infinite, there could still be an "everthing" that doesn't exist?

Perhaps this is more of a philosophical question.

Of all the replies, only JnWaco has correctly answered the original poster's question. And Chalnoth also looks to be on the same track.

This problem is invariably answered incorrectly by most physicists (even the best), simply because they are not specialists in Set Theory, or to be more succinct, transfinite Set Theory. The fact is this. If the Universe is infinite, it may only be "countably" infinite, or equal in cardinality to Aleph Nought (countably infinite = a denumerably infinite set). However, a countably infinite set (= Aleph Nought) is the "smallest" infinity, and is not necessarily exhaustive. As JWaco mentioned, the set of Even Numbers is infinite, yet it is missing an infinite amount of numbers (specifically, all the Odd numbers). A denumerable infinite Set could contain every countable (ordinal) number...with the exception of the number three "3". It is still infinite, but it does not contain all the numbers (in this case, "3"). In fact, just like the Odds, you could instead remove all the Prime Numbers (which are infinite) from the set of Natural Numbers (N), yet you still are left with an infinite set...all the numbers that aren't Prime.

So again, if the Universe is infinite, with cardinality equal to Aleph Nought, then while it may be infinite, it is NOT NECESSARILY exhaustive. That is to say, it is NOT true that every possibility necessarily exists. While it is NECESSARY that the Universe be infinite in order for there to exist every possibility, it is NOT SUFFICIENT.

However, if the Universe has a cardinality equal to the Continuum (= 2^Aleph Nought), then it is possible that it is exhaustive and that it is possible that everything exists somewhere...as the Original Post questions.

In conclusion, it all comes down to the question: If the Universe is infinite, is it countably infinite (i.e. denumerable, equal in cardinality to the Natural Numbers = Aleph Nought), or is it an Aleph greater then Aleph Nought? Only if the infinite Universe is greater in cardinality then Aleph Nought can there exist the sufficient condition/possibility that everything exists somewhere.

[xantox]

A countably infinite set (= Aleph Nought) is the "smallest" infinity, and is not necessarily exhaustive [..] A denumerable infinite Set could contain every countable (ordinal) number...with the exception of the number three "3". It is still infinite, but it does not contain all the numbers (in this case, "3"). [..] Only if the infinite Universe is greater in cardinality then Aleph Nought can there exist the sufficient condition/possibility that everything exists somewhere. [..] It all comes down to the question: If the Universe is infinite, is it countably infinite (i.e. denumerable, equal in cardinality to the Natural Numbers = Aleph Nought), or is it an Aleph greater then Aleph Nought?
An uncountably infinite set is also not necessarily "exhaustive", eg it could also not contain "3".

This problem is invariably answered incorrectly by most physicists (even the best), simply because they are not specialists in Set Theory
Instead of assuming that physicists don't know about set theory, consider that they may take into account the additional constraints of the full physical theories – this must be the case when attempting to answer a physical question, set theory alone won't be sufficient to answer it. Eg if cosmological inflation is assumed, then a condition of ergodicity and randomness could apply on the initial conditions of an infinite universe. In that case, and adding to this that the number of states in a finite volume at finite temperature is also finite, all that can exist physically [1] and within certain temperature limits would exist somewhere.
___
[1] the question is implicitly about physical existence – it would be probably meaningless to require that physical "exhaustivity" should include unphysical states.

[Deuterium2H]

Xantox,

I never stated that an uncountably infinite set was necessarily exhaustive. I stated that it was possibly exhaustive. I specifically used the example of an uncountably infinite set
(e.g. the cardinality of the Continuum) to contrast it with a countably infinite set. And I explained that if the Universe was only countably infinite, that it was a necessary but NOT sufficient condition that "everything exists somewhere". I then provided an example.

Now I agree with you that an uncountably infinite Set may also not be exhaustive. For example, the Set of Real Numbers in the interval [0,1] is uncountable, but not exhaustive. This set also does not contain the number "3". Nevertheless, it is also the case that an uncountable Set of the same Cardinality (2^Aleph Nought) may be exhaustive.
For example the Power Set of |N| = P(N) is the Set of ALL subsets of the Natural numbers, and thus definitely does contain the number "3". Things get very tricky when dealing with transfinite Sets.

Finally, I respectfully make the comment that your citation of "unphysical" states has no meaning in Cosmology. By definition, the Universe contains everything that is physical, and nothing that is non physical. I presume that by your term "unphysical" you technicall mean non-physical. While I agree that non-physical states arise as mathematical constructs in Quantum Field Theory and String Theory, these non-physical states are eliminated by employing gauge symmetry methods. In any event, your example of a finite phase space ("the number of states in a finite volume at finite temperature is also finite") is irrelevent for two reasons. Firstly, because a phase space can also be infinite. Secondly, and more importantly, the very subject of this topic/original post posits that the Universe is Infinite.

[Godswitch]

Surely there are exotic elements we know nothing about that would allow for seemingly improbable situations :wink:

[Deuterium2H]

Surely there are exotic elements we know nothing about that would allow for seemingly improbable situations :wink:

Even so, this still would not make it a necessary and sufficient condition for "everything to exist somewhere".

[xantox]

Xantox, I never stated that an uncountably infinite set was necessarily exhaustive. I stated that it was possibly exhaustive.
Yes, but how "exhaustivity" is defined here? The set of all real numbers does contain all real numbers. But it does not contain complex numbers. Is it "exhaustive" then? To define exhaustivity we should also define the space of states. If it is the integers, then the set of all integers is countably infinite and exhaustive. A dice has only 6 states. We can say in probability theory that the 6 outcomes of a rolling dice are collectively exhaustive.

Finally, I respectfully make the comment that your citation of "unphysical" states has no meaning in Cosmology. By definition, the Universe contains everything that is physical, and nothing that is non physical. I presume that by your term "unphysical" you technicall mean non-physical.
The term "unphysical" is the one most commonly used in the literature – see http://arxiv.org/find/all/1/all:+unphysical/0/1/0/all/0/1 for some usage. Indeed it is just a synonym for "non physical". No big deal anyway on which spelling we use. An unphysical state is something we can come up mathematically but that is against the laws of physics. Like traveling faster than the speed of light. So that it has probably no meaning to require that for "everything to exist" we need to include things that would travel faster than the speed of light. Once we exclude all unphysical states, what remains can be well only countably infinite.

In any event, your example of a finite phase space ("the number of states in a finite volume at finite temperature is also finite") is irrelevent for two reasons. Firstly, because a phase space can also be infinite. Secondly, and more importantly, the very subject of this topic/original post posits that the Universe is Infinite.
The meaning of saying that the number of states of finite volume at finite temperature is finite, is that as a consequence, the number of states of an universe behaving that way, when we assume it to be infinite, is countably infinite.

[Chalnoth]

Yes, but how "exhaustivity" is defined here? The set of all real numbers does contain all real numbers. But it does not contain complex numbers. Is it "exhaustive" then? To define exhaustivity we should also define the space of states. If it is the integers, then the set of all integers is countably infinite and exhaustive. A dice has only 6 states. We can say in probability theory that the 6 outcomes of a rolling dice are collectively exhaustive.
One interesting thing is that an infinite subset of all integers is also exhaustive, such as, for instance, the set of all even integers (because the set of all even integers has a one-to-one relationship with the set of all integers, it is equivalent to the set of all integers).

The term "unphysical" is the one most commonly used in the literature – see http://arxiv.org/find/all/1/all:+unphysical/0/1/0/all/0/1 for some usage. Indeed it is just a synonym for "non physical". No big deal anyway on which spelling we use. An unphysical state is something we can come up mathematically but that is against the laws of physics. Like traveling faster than the speed of light. So that it has probably no meaning to require that for "everything to exist" we need to include things that would travel faster than the speed of light. Once we exclude all unphysical states, what remains can be well only countably infinite.
I strongly suspect that an actual TOE would include no unphysical states.

[Deuterium2H]

One interesting thing is that an infinite subset of all integers is also exhaustive, such as, for instance, the set of all even integers (because the set of all even integers has a one-to-one relationship with the set of all integers, it is equivalent to the set of all integers).
.

Woops, a bit of clarification is required, here, Chalnoth. You are correct that the Set of all Even, natural numbers has the same size (i.e. cardinality) as the Set of all natural numbers (N)...however, the two sets are not "equal", in the sense that they do not contain identical members. The Set of even natural numbers does not exhaust all the natural numbers. However, taking the Power Set of N would ensure that you exhaust all the Natural numbers.

[Deuterium2H]

The meaning of saying that the number of states of finite volume at finite temperature is finite, is that as a consequence, the number of states of an universe behaving that way, when we assume it to be infinite, is countably infinite.

Not necessarily. What if the number of finite volumes in the Universe is itself uncountable. Then, the Universe would be uncountably infinite.

[Chalnoth]

Woops, a bit of clarification is required, here, Chalnoth. You are correct that the Set of all Even, natural numbers has the same size (i.e. cardinality) as the Set of all natural numbers (N)...however, the two sets are not "equal", in the sense that they do not contain identical members. The Set of even natural numbers does not exhaust all the natural numbers. However, taking the Power Set of N would ensure that you exhaust all the Natural numbers.
If the two sets have a one-to-one correspondence, however, the two sets are identical in every way. That is, in any sort of mathematical structure where I use the set of all natural numbers, I can also use the set of all even numbers and everything will always work out the same, as long as I carry through the effects of that correspondence.

[Deuterium2H]

If the two sets have a one-to-one correspondence, however, the two sets are identical in every way. That is, in any sort of mathematical structure where I use the set of all natural numbers, I can also use the set of all even numbers and everything will always work out the same, as long as I carry through the effects of that correspondence.

Hi Chalnoth,

I must disagree. I believe you are confusing equivalence in Set Cardinality with Set equality. While two sets may have the same Cardinality, they are not necessarily equal. For example, take the finite sets X = {1,a,3,4,5}
and the set Y = {1,2,3,4,5}.

The two sets are equal in cardinality. That is |X| = |Y|...where |X| stands for the cardinality of set X. Both sets are equipotent.

However, the sets are not equal...that is, X does not equal Y, because set X has the member "a" whereas set Y has a member "2".

By the definition of Sets, two Sets are equal if and only if they have the same elements.

The Set of Rational numbers has the same cardinality as the set of Natural numbers. Both sets have a Cardinality = Aleph Nought. However, try as one might, you will never find the element "1/3" in the Set of Natural Numbers. The two sets are not equal in membership, although the are "equal" in size. Technically, one can only use the equality sign when comparing the cardinality of these sets, i.e.:
|Q| = |N| is a true statement. However, {Q} = {N} is NOT a true statement.

Another example would be the Set of Algebraic Numbers. They can be put in a one-to-one correspondence, and thus have the same Cardinality as the Natural Numbers. In fact, the Set of Natural numbers is a proper subset of the Set of Algebraic Numbers, even though they are equal in size/cardinality. However, if one were tasked to pick out squareroot(2) from the Set of Natural numbers, one would be at a loss.

[Chalnoth]

Hi Chalnoth,

I must disagree. I believe you are confusing equivalence in Set Cardinality with Set equality. While two sets may have the same Cardinality, they are not necessarily equal. For example, take the finite sets X = {1,a,3,4,5}
and the set Y = {1,2,3,4,5}.

The two sets are equal in cardinality. That is |X| = |Y|...where |X| stands for the cardinality of set X. Both sets are equipotent.

However, the sets are not equal...that is, X does not equal Y, because set X has the member "a" whereas set Y has a member "2".

By the definition of Sets, two Sets are equal if and only if they have the same elements.

The Set of Rational numbers has the same cardinality as the set of Natural numbers. Both sets have a Cardinality = Aleph Nought. However, try as one might, you will never find the element "1/3" in the Set of Natural Numbers. The two sets are not equal in membership, although the are "equal" in size. Technically, one can only use the equality sign when comparing the cardinality of these sets, i.e.:
|Q| = |N| is a true statement. However, {Q} = {N} is NOT a true statement.

Another example would be the Set of Algebraic Numbers. They can be put in a one-to-one correspondence, and thus have the same Cardinality as the Natural Numbers. In fact, the Set of Natural numbers is a proper subset of the Set of Algebraic Numbers, even though they are equal in size/cardinality. However, if one were tasked to pick out squareroot(2) from the Set of Natural numbers, one would be at a loss.
Well, yes, this is strictly true. But since we're talking about this in the context of a physical law (assuming, for a moment, that we're trying to keep this on the topic of the original post), then set equality is not the proper metric.

Consider in the context of physical law, a set would be one component of the full mathematical structure. Let's imagine, for the sake of argument, that the full mathematical structure we are talking about is an algebra. I can define an algebra with the set of natural numbers combined with addition. If you give me any set that has a one-to-one correspondence with the natural numbers, I can define an algebra in such a way that the behavior of this other set is identical to the behavior of the algebra with natural numbers (though the operator may, depending upon the set, look nothing like addition).

In the end, this doesn't matter for the physics. What we call a specific number is irrelevant. It is only the interrelationships that matter for defining the behavior of the mathematical structure.

[Deuterium2H]

Well, yes, this is strictly true. But since we're talking about this in the context of a physical law (assuming, for a moment, that we're trying to keep this on the topic of the original post), then set equality is not the proper metric.

Consider in the context of physical law, a set would be one component of the full mathematical structure. Let's imagine, for the sake of argument, that the full mathematical structure we are talking about is an algebra. I can define an algebra with the set of natural numbers combined with addition. If you give me any set that has a one-to-one correspondence with the natural numbers, I can define an algebra in such a way that the behavior of this other set is identical to the behavior of the algebra with natural numbers (though the operator may, depending upon the set, look nothing like addition).

In the end, this doesn't matter for the physics. What we call a specific number is irrelevant. It is only the interrelationships that matter for defining the behavior of the mathematical structure.

Fair enough. So getting back on topic, do we both agree, then, that it is not a necessary and sufficient condition that the Universe be infinite in order that "everything exists somewhere". The crux of my argument was to rebut the commonly held belief (even amongst some physicists) that an infinite universe implies that all possibile states exist and that somewhere out there is an exact duplicate of myself typing this very post.

[Chalnoth]

Fair enough. So getting back on topic, do we both agree, then, that it is not a necessary and sufficient condition that the Universe be infinite in order that "everything exists somewhere".
Yes. You also need a physical mechanism to explore all possibilities.

The crux of my argument was to rebut the commonly held belief (even amongst some physicists) that an infinite universe implies that all possibile states exist and that somewhere out there is an exact duplicate of myself typing this very post.
Well, that particular possibility is a necessary consequence of an infinite universe combined with inflation.

[Deuterium2H]

Well, that particular possibility is a necessary consequence of an infinite universe combined with inflation.

Chalnoth...you lost me. Perhaps I misunderstand your statement...however, it seems you are now stating that "everything exists somewhere" as being a necessary consequence of an infinite Universe combined with inflation.

As already discussed, an infinite Universe is not a sufficient condition that "everthing exists somewhere", and inflation does not change this in the least. Inflation is just an exponential expansion. If the Universe was created infinite, then inflation doesn't make it a higher power of infinity...nor does it in any way change what may be a countably infinite Universe into an uncountably infinite Universe.

[xantox]

I strongly suspect that an actual TOE would include no unphysical states.
I believe the same.

Not necessarily. What if the number of finite volumes in the Universe is itself uncountable. Then, the Universe would be uncountably infinite.
Note that you said in your first message "If the Universe is infinite, it may only be "countably" infinite". Anyway, it is possible to map to naturals all permutations of state at finite temperature of any causally connected patch of the universe (observable universe). That is all which may exist, within known law of physics.

So getting back on topic, do we both agree, then, that it is not a necessary and sufficient condition that the Universe be infinite in order that "everything exists somewhere". The crux of my argument was to rebut the commonly held belief (even amongst some physicists) that an infinite universe implies that all possibile states exist and that somewhere out there is an exact duplicate of myself typing this very post.
This part is fine. The problem was on the other part, where you said that a necessary condition for all possible states to exist is that "the universe must be infinitely uncountable". I don't see any reason for that to be true.

[Chalnoth]

Chalnoth...you lost me. Perhaps I misunderstand your statement...however, it seems you are now stating that "everything exists somewhere" as being a necessary consequence of an infinite Universe combined with inflation.
Not quite. Now, for the sake of argument, I think it would be better to think of eternal inflation instead of "infinite universe plus inflation". They have the same implications here anyway, except that eternal inflation is at least somewhat more reasonable as a physical theory.

With that out of the way, the argument here isn't that everything happens somewhere, but that eternal inflation is a mechanism for exploring some subset of the possible parameter space. Now, you can actually calculate that the possible configurations for a universe like our own is quite finite. And since eternal inflation produces an infinite number of Hubble volumes, and those Hubble volumes form a finite set of possible configurations, any configuration that eternal inflation explores once will be explored an infinite number of times.

[Deuterium2H]

I believe the same.


This part is fine. The problem was on the other part, where you said that a necessary condition for all possible states to exist is that "the universe must be infinitely uncountable". I don't see any reason for that to be true.

?? Didn't I say necessary but NOT sufficient. Certainly, the Universe has to be infinite (that is a necessary condition) for there to exist the possibility that "everything exists somewhere." But that in and of itself does not make it a sufficient condition. That is what I have been arguing all along. I don't think I had stated anywhere that an uncountably infinite Universe was a necessary AND sufficient condition for this to occur...if I did, that was an unintentional mistake.

[xantox]

?? Didn't I say necessary but NOT sufficient. Certainly, the Universe has to be infinite (that is a necessary condition) for there to exist the possibility that "everything exists somewhere." But that in and of itself does not make it a sufficient condition. That is what I have been arguing all along. I don't think I had stated anywhere that an uncountably infinite Universe was a necessary AND sufficient condition for this to occur...if I did, that was an unintentional mistake.

Yes, but the problem is that it is not even a necessary condition. A countably infinite universe can be exhaustive, too.

[Sage Lee]

Ok - first of all, I apologize for what is likey to be a very long-winded example of my rambulitis.

It will soon be clear that I know absolutely nothing about any of this; I can hardly follow half of the jargon that you guys throw around so casually. I only came across this thread (and forum) by googling the question that is in the thread title, because I'm just crazy like that and found myself thinking about infinity (again), and I wanted to hear some smart-people thoughts on the matter.

But I quickly found myself over my head. I don't know what Hubble volume is; I don't know what TOE stands for, I don't really know what the Copenhagen interpretation is (although I'm sure I've read all about these concepts on Wikipedia at some point or another, because that's just what I do.) I suppose I could go and refresh my Wiki knowledge (and I probably will, sigh), but I know that if I try I will inevitably find something I don't understand within the explantion of what I'm trying to understand, which will lead me to delve into an explantion of that, which will of course contain another term or concept I don't understand, and so on, until I have 50 pages of advanced physics concepts opened on my web browser and a throbbing mental headache. The problem lies in the fact that there probably aren't too many laymen that are interested in discussing the finer points of such complicated topics, but there's at least one (hai dere!) So basically, what I'm trying to say is: be gentle.

So, all these different interpretations of infinity, countable and uncountable, etc etc... these just seems like different ways of putting a limit on infinity, which by (my) definition should have no limits. For instance, the example of how a set containing only even numbers could be infinite and yet not exhuastive... that was a great explanation, but it still seems to me that a finite limit has been put on the (my) basic concept of infinity. It's like saying an "infinite line"... to me that seems like a misnomer, simply because the phrase itself puts a finite parameter (a line) on infinity. Put another way, it's like saying infinity, but in only one direction. Which (to me) means it's not actually "infinite" at all, it just happens to go on forever in that one direction.

In my mind, imagining infinity (ha!) is more like picturing a sphere that expands outwards in all directions and never stops. In fact, time itself is kind of like this infinite line I mentioned, and by existing in the first place it already tells my feeble brain that a true infinity isn't possible in our observable universe. If infinity truly existed, physically, it seems to me that it would be everything, everywhere, EVER... happening all at once (and everywhere at once.) Over and over and over again, until my head assploded.

I'm realizing now that my defintion of infinity (everything) is the exact opposite of the definition of zero (nothing). I don't know if this is intuitive or if there's some mathematical basis for that, or if it's simply just incorrect, but that's how I've always defined infinity: on a number line, it's the polar opposite of zero, and to extend that concept in a philisophical sense is to make it the polar opposite of nothing.

But let's assume that we're only talking about infinite physical space. Time, whether I like it or not, seems to exist, even if only to keep everything from happening at once. So with this one boundary in place (time), let's assume that physical space goes on forever. I've always taken to heart the concept of "the closer you get to infinity, the probability of x happening approaches 1." And by extension, if you actually could get to infinity, then the probability of x happening, somewhere, sometime, must equal 1.

And I still just can't get past this. How is this not true? What exactly am I missing about this concept of infinity? Using that one example along the lines of "different blurry versions of myself that all slightly vary outwards from point A (the "real" me, from now) and some get hit by the car, or meet the girl, and some don't, blah blah blah" but then you assume this has been going on since the beginning of the universe (or dare I say, since even before that? Maybe it's been going on forever? Maybe the universe itself has infinite variations, an infinite amount of which evolved life similar to ours, or nothing like ours, and likewise, an infinite amount of universes that never were, so to speak.)

Here though, I must clarify once again that when I say infinity, I'm talking about something that all variations are encompassed within. I'm aware of the many-worlds theory, but in my definition of infinity, every world (or dimension, or variation, or whatever) is included within that term. I guess I'm saying that if infinity exists in any real sense, everything that exists, wherever it may be, is contained within that infinity. It's impossible for anything to exist outside it... well, because there is no "outside," it goes on forever, durrr.

So I can't help but stand by the concept that if our reality were infinite, everything would be happening within it. Everything meaning anything that any of us can think of, along with an infinite amount of things we could never possibly think of. And I just don't understand why it's assumed that all reality, even if infinite, would have to conform to our known laws. I don't understand why it's assumed that everything was once connected, as someone put it, to our reality (or something to that effect) and therefore must follow the laws we (think we) know. I mean, from what I understand, there's already contradictions in the "rules" when we try to relate them to very very small or very very large objects (the so far fruitless search for a unified theory), so it follows, for me, that our rules might conceivably not apply once we go even bigger (or smaller.) And when we're talking infinitley bigger (or smaller), well, it seems like everything we (think we) know could be up in the air.

I remember first thinking about this when I was about 14 (I'm 32). I read some sci-fi book that touched on the concept of "everything must exist within infinity", and I thought about it for a long while. The concept just made sense, and it still does, which is why I can't get past this. At the time though, I "proved" to myself that infinty can't exist. I did this by thinking of something that *should* exist, but obviously didn't. I thought of a planet full of alternate "me's" (an infinite amount of them). I then thought of a planet full of "me's" that had found a way to bend time and space and traverse dimensions with but a thought. I then thought of one of these "me's" that could observe (the real) me, and had the power to appear before me, and make himself known to me, and then I thought of a "me" who chose to do just that. And since I never appeared before myself, I thought I had proved that infinity didn't exist. (I then realized that there would be an infinite amount of these me's who would appear before me, as well as an infinite amount of anyone else, and everything else, appearing in front of anyone and everything else, and so on, and that's when I decided that true infinity would mean an unimaginable blur of everything happening all at once, everywhere at once.)

I am now old enough to undertand that the only thing I "proved" is that I didn't understand what the hell I was talking about. But the problem is that I still don't understand, because everything I just said still makes perfect sense to me. Even if we're in finite space, even if time is the only thing that's infinite, it seems to me that sometime, everything I can think of (and everything I can't) must exist, eventually. But again, I'm thinking of a "me" from the future who has figured out how to travel back here to my time, and of course there are an infinte amount of them at some point in the timeline, all of whom can travel back to this exact moment and have the power to make themselves observable to me, and... boom, everything at once.

Anyway, so I guess it comes back to these "rules" or "laws" that we have observed, and whether or not they can ever be broken, given infinity. Can infinty be seperated into sections that can't ever co-exist? I contend that it can't; eventually they must (or already have.) Eventually, given infinity, all of our rules must be broken. So a *true* inifinty cannot exist in any physical sense.

Anyway, sorry for all that. I can never be concise in things like this, for 2 reasons: 1) I don't understand enough of the technical jargon to properly sum up complex thoughts with one or two terms, and 2) I have no idea what I'm talking about.

I guess I'll sum it up my questions here at the bottom for those not inclined to read this whole thing:

Can someone explain to me, as you would to a child, why an infinite universe "isn't sufficient" for *everything* existing? By the same token, why would an infinite timeline be insufficient for everything existing, eventually? Why can't laws (traveling back through time, or across dimensions, and all the rest) be broken, given infinite time or space? Why can't a four-sided triangle exist just because I can't conceptualize it? In infinity, even that should be there somewhere, even if our feeble, logical minds would snap if they ever actually tried to understand it. (To be clear, my whole argument is that these things don't exist, but only because infinity doesn't either, at least beyond a theoretical concept.)

But on that same note, is it possible that logic itself is only a limitation of the human perspective, rather than some universal, infallible ideal? (This is a question I asked someone in another forum recently, where a bunch of people got to arguing about whether or not God exists (and on a poker forum, believe it or not.) One guy (basically) said "No, because [too many things about that] are not logical." Which got me thinking about illogicalities, and the possible limits of human thinking/perception, and about how if God does exist, he could pretty much violate any rule we can think of, because let's face it, he's God. A bit off topic here, but I'm just reiterating the concept of "just because it doesn't make sense to us doesn't mean it's not true.")

In closing, can I just say I ****ing hate v-bulletin? To my great consternation, I swear that everybody uses it now. Someone hurry up and write something better.

[Chalnoth]

Can someone explain to me, as you would to a child, why an infinite universe "isn't sufficient" for *everything* existing? By the same token, why would an infinite timeline be insufficient for everything existing, eventually?
Well, consider a simple case: a list of numbers. If the list of numbers is infinite in length, does this mean every number is represented? Nope. Consider this list:
{1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,...}

The list, in this example, repeats the numbers 1-5 an infinite number of times, but it still only includes the numbers 1-5. The same sort of thing could potentially be the case with reality where, for whatever reason, it is unable to access certain possible configurations, even if it is infinite in size.

Why can't laws (traveling back through time, or across dimensions, and all the rest) be broken, given infinite time or space?
If we define a law of nature as an accurate description of nature, then by definition it cannot possibly be broken: whatever nature does, an accurate law describes it. Now, the laws which we have discovered are all wrong in some regard, meaning that they don't always describe what nature does. But this is because we don't yet know the full laws of nature.

Why can't a four-sided triangle exist just because I can't conceptualize it?
A triangle is defined as having three sides. So saying a four-sided triangle is the same as saying, "a four-sided, three-sided polygon." It is an improper use of language.

But on that same note, is it possible that logic itself is only a limitation of the human perspective, rather than some universal, infallible ideal? (This is a question I asked someone in another forum recently, where a bunch of people got to arguing about whether or not God exists (and on a poker forum, believe it or not.) One guy (basically) said "No, because [too many things about that] are not logical." Which got me thinking about illogicalities, and the possible limits of human thinking/perception, and about how if God does exist, he could pretty much violate any rule we can think of, because let's face it, he's God. A bit off topic here, but I'm just reiterating the concept of "just because it doesn't make sense to us doesn't mean it's not true.")
Basic logic just assumes one thing: logic is consistent. That is to say, whenever you have a definitive statement, that statement is always either true or false. We may not always know which, but it is always one or the other. By only allowing statements in the logic that are either true or false, the laws of logic that can be derived are absolute and inviolable.

One can potentially consider logics that allow for ambiguous or meaningless statements, but often it is easier to just not allow those statements.

[Deuterium2H]

Ok - first of all, I apologize for what is likey to be a very long-winded example of my rambulitis.

It will soon be clear that I know absolutely nothing about any of this; I can hardly follow half of the jargon that you guys throw around so casually. I only came across this thread (and forum) by googling the question that is in the thread title, because I'm just crazy like that and found myself thinking about infinity (again), and I wanted to hear some smart-people thoughts on the matter.

But I quickly found myself over my head. I don't know what Hubble volume is; I don't know what TOE stands for, I don't really know what the Copenhagen interpretation is (although I'm sure I've read all about these concepts on Wikipedia at some point or another, because that's just what I do.) I suppose I could go and refresh my Wiki knowledge (and I probably will, sigh), but I know that if I try I will inevitably find something I don't understand within the explantion of what I'm trying to understand, which will lead me to delve into an explantion of that, which will of course contain another term or concept I don't understand, and so on, until I have 50 pages of advanced physics concepts opened on my web browser and a throbbing mental headache. The problem lies in the fact that there probably aren't too many laymen that are interested in discussing the finer points of such complicated topics, but there's at least one (hai dere!) So basically, what I'm trying to say is: be gentle.

Sage,

There is absolutely nothing here to be embarrassed or uncomfortable about. In fact, you are in good Company. From at least the time of the ancient Greeks (and most likely much earlier) up until the late 19th Century, mankind has struggled with the the metaphysical and mathematical concept of infinity. In fact, it wasn't until well into the beginning of the 20th century that Georg Cantor's revolutionary work on Set Theory and Transfinite numbers was put on firm axiomatic foundations, and accepted by the mainstream mathematical community. If you can just imagine the breadth of time that has passed since antiquity (3,000 plus years), in which many of the GREATEST mathematical minds in history struggled with the seemingly paradoxical characteristics of the infinite, then this fact should humble us all.

Just to add a bit more context to the problem of infinity represents what is now called one of the Great "crisis" in Mathematics. And in a way, the concept of infinity was directly or indirectly involved in each great crisis.

The first great "crisis" was the discovery, by the Greeks, of the Irrational Numbers. How this came to be, and how they dealt with them (or perhaps more aptly put, ignored them), entire books have been written. The theory of Irrational numbers is intimately tied up with the Theory of Real Numbers, which in itself is intimately tied up with Set Theory, and the concept of completed, infinite Sets.

The second great "crisis" involved the fact that the development of the Calculus had no rigourous foundations, even though Newton and Liebniz's methods worked, and solved previously intractable physical problems. Key to both Newton's and Liebniz's Calculus was the concept of infinitesmals, as well as the approach to a Limit. Both are inexorably wrapped up with the concept of infinity. It wasn't until Cauchy, Bolzano and Weierstrass (in the early 1800's) that Calculus was put more or less on a firm foundation...despite the fact that there as yet existed no rigorous foundation for the Real Numbers (and, by consequence, Irrationals, Rationals, and even the Natural Numbers).

The third "crisis" involved the "discovery" and development of Non-Euclidean Geometry, by Gauss, Riemann, and others. Again, the Infinite reared it's head, as non-Euclidean geometries were predicated upon assuming the falsification of Euclid's fifth postulate (parallel line postulate).

The last great "crisis" involved the very foundations of Mathematics, and at it's very heart was the development of Set Theory and Transfinite numbers. Again, entire books have been written on this topic. Suffice it to say that Cantor's Set Theory and transfinite numbers shook the very pillars of mathematics, and eventually led to Godel's Incompleteness Theorem(s), which set limits on what was trully "knowable" in mathematics. In short, within a given mathematical system, certain logical statements can neither be proved nor disproved.


So, all these different interpretations of infinity, countable and uncountable, etc etc... these just seems like different ways of putting a limit on infinity, which by (my) definition should have no limits. For instance, the example of how a set containing only even numbers could be infinite and yet not exhuastive... that was a great explanation, but it still seems to me that a finite limit has been put on the (my) basic concept of infinity. It's like saying an "infinite line"... to me that seems like a misnomer, simply because the phrase itself puts a finite parameter (a line) on infinity. Put another way, it's like saying infinity, but in only one direction. Which (to me) means it's not actually "infinite" at all, it just happens to go on forever in that one direction.

Sage, you may be mixing up two concepts...that of a Line, and that of a Line Segment. By it's very nature, a Line (in the strict geometric sense) is infinite in length. A Line Segment is bounded, and of finite length. A line that starts at a point, and goes on forever in one direction is just as infinite as one that goes in both directions. When dealing with Infinity, our natural intuition is of no help...and in fact only get's us in trouble. As an example, I just previously claimed that a line segment is finite. And in one sense, it is, in that it is both bounded and has a definite, finite extent. However, that same "finite" line segment is composed of an infinite number of points. For those unfamiliar with Set Theory, it comes as a real shock to learn that there are EXACTLY the same number of points on the line interval from [0,1] as there are on an interval twice as long [0,2]. No more, no less. In fact, there are the same number of points. In math-speak, we say that there is a one-to-one correspondence between the set of Real numbers in the interval [0,1] and the interval [0,2]. How can we prove this? We establish a Function that maps each and every Real number in the smaller interval with those in the larger interval. That function would be:
y = f(x) = 2x

That is to say, take any Real number "x" in [0,1], and double it, via the the function f(x) = 2x. The result is that you will have paired of each Real number in the smaller interval with exactly one Real number in the larger interval. Technically, this is called a bijection, which is "one-to-one" and "onto". When dealing with infinite sets, the phrase "the whole is always greater then one of it's parts" is no longer valid. In fact, the very definition of a infinity (i.e. an Infinite Set) is any Set that can be put in a one-to-one correspondence with at least one of it's proper Subsets. Another example would be the Set of all Natural Numbers and a proper Subset of just the Even Numbers. Both of these Sets contain exactly the same number of members, and are the same "size" (otherwise known as Cardinality). We know this because we can "count" by making a one-to-one correspondence between each Natural number and each Even number, like so:

1 -> 2
2 -> 4
3 -> 6
4 -> 8
5 -> 10

Each Natural Number is matched with exactly one Even number, and vice versa.


In my mind, imagining infinity (ha!) is more like picturing a sphere that expands outwards in all directions and never stops. In fact, time itself is kind of like this infinite line I mentioned, and by existing in the first place it already tells my feeble brain that a true infinity isn't possible in our observable universe. If infinity truly existed, physically, it seems to me that it would be everything, everywhere, EVER... happening all at once (and everywhere at once.) Over and over and over again, until my head assploded.

What you just described happens to be one of the great stumbling blocks in the mathematical history of Infinity. Just as you described a sphere that expands outwards in all direction, and never stops, is exactly how pre-Cantorian mathematicians conceived infinity. They only accepted a "potential" infinity. A potential infinity was any process that could be continued indefinitely, and never ends or completes, such as the sequence of numbers: 1, 2, 3, 4, 5...
An actual or "completed" infinity is thinking of those same numbers, but taken as a complete, single Set, i.e.: {1,2,3,4,5...}
"A set is a many that allows itself to be thought of as a one."
The difference between a "potential" and an "actual" infinity may seem subtle, but it lies at the core of modern mathematics. Once infinite sets are taken as completed wholes, they can be manipulated and worked with.

Perhaps the single biggest surprise, when first learning transfinite Set theory, is that not all infinite Sets are equal. That is to say, there exists larger sizes of infinity. The smallest infinite Set is the Set of Natural Numbers, which is equal in size to the Set of Integers, which is equal in size to the Set of Rational Numbers. They all are equal in size, and all of the aforementioned numbers comprise the smallest Infinity, also called a "countable" or "denumerable" infinity, and all are designated by the Cardinal number Aleph-Nought. It is quite counter-intuitive to think that the Set of Rational Numbers is no greater in size then the counting numbers...especially when you consider that between any two Natural numbers (e.g. number "2" and number "3") there are an infinite number of Rational numbers. Futhermore, between any two Rational numbers there are an infinite amount of more Rational numbers. Yet, the number of Rationals is exactly the same as the number of Naturals. The Set of Natural numbers is bijective with, and can be put in a one-to-one correspondence with the Set of Integers, the Set of Rationals, and even the Set of Algebraic Irrationals.

As mentioned previously, there exists greater Infinite sets (in fact, an infinite number). The Set of Real numbers is one example. It's size is greater then the Natural/Integer/Rational numbers. Mathematically, the Set of Real numbers = ( 2 ^ |N| )...where |N| is the Cardinality (size) of the Natural numbers. Another way of stating this is that the Set of Reals is equal to the Set of ALL Subsets of Natural Numbers.



Can someone explain to me, as you would to a child, why an infinite universe "isn't sufficient" for *everything* existing? By the same token, why would an infinite timeline be insufficient for everything existing, eventually? Why can't laws (traveling back through time, or across dimensions, and all the rest) be broken, given infinite time or space? Why can't a four-sided triangle exist just because I can't conceptualize it? In infinity, even that should be there somewhere, even if our feeble, logical minds would snap if they ever actually tried to understand it. (To be clear, my whole argument is that these things don't exist, but only because infinity doesn't either, at least beyond a theoretical concept.)



In order to understand this, you need to understand the formal, logical distinction between what is a "necessary" condition, and what is a "necessary AND sufficient" condition. They are not the same. I guess the best way to explain is through an analogy and example.
To say that X is a necessary condition for Y is to say that it is impossible to have Y without X. In other words, the absence of X guarantees the absence of Y.
Example: Having four sides is a Necessary condition for being a Square.
Notice, however, it is not a Sufficient condition. For example, a Rectangle has four sides, as does a Rhombus, but they are not necessarily Squares. A Rectangle has four equal angles, but may not have four equal sides. Conversely, a Rhombus has four equal sides, but may not have four equal angles.
Compare/contrast the above example to the following:
A quadrilateral with four equal sides and four equal angles is a both Necessary and Sufficient condition for being a Square.
-Or- another way of phrasing this: A quadrilateral that is BOTH a Rectangle AND a Rhobus is a Necessary and Sufficient Condition for being a Square.

Now, getting back to your question as to how an Infinite Universe isn't a "Sufficient" condition for "Everything existing somewhere"...
It is a Necessary condition that the Universe be Infinite in order for there to exist the possibilty that "everything exists somewhere". This is obviously trivially true, because if it were not infinite, then it would be finite, and a finite Universe cannot be a Necessary condition for everything existing somewhere. So, as a minimum, it is a Necessary condition that the Universe be Infinite in order for this possibility to exist. However, that is not a Sufficient condition. As discussed in earlier posts in this thread, the Universe may be "countably" infinite...that is to say, having the same size (Cardinality) as the countably Infinite Set of Natural Numbers ( |N| ). However, the Set of all Even numbers is just as big (i.e. the same size) as the Set of all Natural Numbers, yet the former Set is missing an infinite amount of numbers...that is, the Odd numbers. So, these two sets have exactly the same NUMBER of elements (members), but these two sets are not "identical", and only one of these sets "exhaust" all the Natural numbers, whereas the other set does not.

With that said, I am not exactly certain what would be both a Necessary and Sufficient condition for an infinite Universe to ensure that "everything exists somewhere". From a purely mathematical perspective, I might argue that the Universe would need to have the Cardinality of the Continuum (= the Set of Real numbers). However, one could equally argue that that, in and of itself, may not even be a Sufficient condition. The tiny interval [0,1] on the Real number line is everywhere Dense and Continuous, and this segment contains an equal number of points as in the entire Real Number line. In fact, it contains in equal number of points as on a plane. Moreover, it contains just as many points as on any finite n-dimensional space. Nevertheless, despite the equipollence of the interval [0,1] with the entire Real Number line, it is not "exhaustive". It doesn't contain the number "2", or "pi", or "e", or for that matter any Real number greater then one or less then zero.

All this gobbledygook ultimately comes down to the conclusion that, even though the Universe may be infinite, it does not necessarily follow that "everything exists somewhere".

[Sage Lee]

Well, consider a simple case: a list of numbers. If the list of numbers is infinite in length, does this mean every number is represented? Nope. Consider this list:
{1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,...}

The list, in this example, repeats the numbers 1-5 an infinite number of times, but it still only includes the numbers 1-5. The same sort of thing could potentially be the case with reality where, for whatever reason, it is unable to access certain possible configurations, even if it is infinite in size.

Okay... honestly, I kind of rolled my eyes (at first) when I saw that you just rehashed the same example as had been stated previously, using a finite list of numbers (except this time you said "1 through 5" instead of "only even numbers". Because this is where I suffer a disconnect: you're saying, it's infinite!!!!..... eeeeexcept it stops at 5. To my way of thinking, saying it stops at 5 is already cheating, because if it stops at 1 on one end and 5 on the other end, it's not really infinite, is it? It's limited and therefore in a sense *finite* in that it can only use five numbers.

If we were to use numbers to represent an infinite reality, I would've thought that we must by definition have no limits on the numbers we choose to use, if we're talking about infinite. Like, an infinite representation using only numbers would by definition have to include all numbers, positive and negative, odd and even, real and imaginary, all integers and complex numbers and everything in between; it would go on forever in all directions, with no finite "bookends," so to speak (1 and 5). And there would be an infinite amount of 1's and 2's and 6's and 10's and an infinite amount of each negative number and imaginary number and so on... or else it wouldn't be an accurate representation of "infinite."

However, funnily enough, your italicized "could" made me realize what you're trying to say more than the reiteration of the example itself: "just because the universe is infinite in one sense doesn't mean it's infinite in all senses." ("Sense" of course isn't the right word here, but you probably get what I'm trying to say. Perhaps it would be better to say that "just because the universe is infinite along one dimension (space) doesn't mean it's infinite along infinite dimensions...?") I think this is what you're saying, although you use the word "exhaustive" instead of "infinite along infinite dimensions," when all along I've used "infinity" synonymously with "exhaustive" or "without any limits at all."

So I can see that if we're only talking about being infinite along one dimension or whatever, then infinite space, for instance, is indeed not enough to imply an all-inclusive set. So what would be sufficient for all things (and non-things) to exist? What kind of infinity am I thinking of that implies supreme inclusivity? Is there a term to address this concept (because "infinity" obviously doesn't quite cut it), or am I just wading in too-murky waters here? Are we talking about infinite space and infinite time and infinite ? Infinite infinitivity? Okay, that sounds dumb, I'm just trying to be clear because I'm not so sure that I'm being clear at all.

A triangle is [i]defined as having three sides. So saying a four-sided triangle is the same as saying, "a four-sided, three-sided polygon." It is an improper use of language.

Yes, yes, of course... believe me, I do understand this. But there is still a part of me that wants to say "just because my puny mind can't comprehend the existence of something that has only three sides while still *somehow* having four sides doesn't mean that it's not possible." I mean, I don't honestly think it is possible, but as I said, a part of me romanticizes that this could simply be a function of the human mind's inability to think outside the proverbial box rather than a testament to the supreme infallibility of logic. Which is why I joked that we would go mad if we ever actually comprehended a "three sided and yet four sided" thing, to get across the idea that it's possible (although extremely unlikely) that such a thing could actually exist whether or not we understand it.

But I do of course understand what you are saying, which is basically summed up here:

Basic logic just assumes one thing: logic is consistent. That is to say, whenever you have a definitive statement, that statement is always either true or false. We may not always know which, but it is always one or the other. By only allowing statements in the logic that are either true or false, the laws of logic that can be derived are absolute and inviolable.

One can potentially consider logics that allow for ambiguous or meaningless statements, but often it is easier to just not allow those statements.

And yet the bolded has always bothered me, because yes it is easier but not necessarily correct. The fact remains that a contradiction can't really exist... except, of course, by some kind of magic or supreme omnipotence beyond my ability to understand. Which is kind of what I was getting at. I always think of these things in the context of a supreme omnipotence - if there was a supremely omnipotent God, could he draw a square circle? Could he make a burrito so hot that even he couldn't eat it? Of course not, that doesn't make any sense... except that maybe - just maybe - he could. Because, duh, he's frickin God isn't he? He could, theoretically, create a reality that we perceive, that seems to behave in a certain way but that isn't at all indicative of how things might actually be outside our sphere of observation.

But whatever, I do understand that it's kind of pointless to talk about things in such a way, we can only use what we have (or what we can observe, or what we can comprehend.) It's just easy for me to talk like this considering that there is just so much that has been shown to be incorrect as our observational capabilities have grown that it's hard for me to accept that anything at all is set in stone. I mean, I even recall recently reading an article about a paper by somebody or another postulating that gravity doesn't really exist. It was full of concepts and equations that I don't know enough about to properly ponder, so I didn't really try, and I guess the whole idea has gotten resistance from some other smart people, but I can only assume the original writer of the paper is pretty smart too and is convinced of the work, so I guess only time will tell if he/she/they can prove their thoughts or not. But in this sense, who knows what might be proven as a falsehood, given enough time?

Anyway, it seems to me this whole thing does indeed prove that the universe isn't infinite in totality; it isn't infinite along all dimensions or whatever (but again, I don't really know how to properly say what I'm trying to say here) or else I'd have an infinite amount of past and future and present "me's" (and an infinite amount of everything else) occupying the entirety of an infinite amount of space. To use the example of a line again, a line running east to west that goes on forever will never, ever, go north or south. Nor will it ever go up or down. It can never escape its own boundaries of being just a flat, unbending line, and I have trouble with infiinity being used in the context of something that has such obviously finite boundaries.

So reality might be spatially infinite, but that doesn't mean it's not finite in the sense that it's still limited to certain configurations (only 1 through 5; only east to west.) I would still ask though, if there is a term to properly describe this concept of an all-inclusive infinity, because I have a feeling that "infinite along all dimensions" isn't really saying what I mean to say; I can only hope you understand what I'm trying to get at. Is "exhaustive" all we have for that? Maybe I could say then that while the universe may be infinite, it's not possible for the universe to be exhaustive, or else everything would exist all at once in some unimaginable blur of... well, everything at once. Would this be correct, and if so, is there a better way to say it? If incorrect, what assumptions am I making here that I shouldn't be?

I actually came up with my own term for an "all-inclusive infinity" a long time ago when I was trying to *prove* a theory (again using this term loosely since I'm a half-wit in these matters) that reality doesn't need an observer to exist on it's own. (In other words, that reality can exist whether or not God is watching, because I heard that some very smart physicists were beginning to think that he or someone must be observing or else we wouldn't exist, and that got me thinking.) I don't remember what the term was, but it sounded cool. Dimensional Infinitum, or something like that. Forgive me, I tend to pull these things out of my ***.

Anyway, the whole idea that something needs to be observed in order to exist has never sat well with me, so I came up with the aforementioned and half-baked theory one time when I was quite ill and admittedley feeling a bit loopy; I lovingly refer to this theory as "Masturbational Existentiality."

***Much of what follows will likely be nonsense, so read on at your own risk; however, I feel compelled to share this simply because I can and because no one I know would ever humour (much less understand) me. But again, forgive me for being such an amateur and for my illusions of grandeur. (Plus my computer crashed a while back and I lost all of the nonsense I'd written on the subject, so this is all from memory and as such, probably a bit more wishy-washy than I would hope.) I can only hope that what I'm about to say is at least entertaining, in some fashion or another.***

My theory - M.E. for short - postulated and attempted to prove, among other things, that:

1) Reality *is* whether or not anyone *else* is watching (measuring/observing, whatever)
2) In order to exist in observable reality, something must be capable of observing itself
3) Particles have free will, and so does a tomato
4) Reality is finite in some sense, and because of this, everything in reality could be defined in terms of a tomato

The tomato thing is intentionally ludicrous, but this is all, of course, tongue-in-cheek, or I wouldn't call it Masturbational Existentiality. (The name is taken from the fact that *if* some kind of observance is indeed necessary in order to exist, and *if* something can exist all on it's own, without any outside observation, then something that exists must be capable of observing (interacting with) itself... interacting with itself, get it? Masturbational Existentiality. Also, I'm essentially stroking my own ego by even pretending to think competently about things like this, so there's a double meaning there: I'm stroking myself. I'm sorry, but I still find all of this funny and yet deadly serious at the same time. ZOMG an existing contradiction!!!!)

I used the following and quite logical statement to *prove* that no outside observation is necessary for reality to exist, all on it's own:

If there is a rock, than there is a rock.

I still laugh at myself every time I write this, because it still seems quite inarguable while still of course being nonsense. I mean, if there's a rock, then there is indeed a rock, right? Conversely, if there is not a rock, then there is not a rock; it's still true both ways, which, if I recall, is important when dealing with logical statements. I've no doubt that anyone who's fluent in logic will gladly inform me that there is some name for this type of ridiculously stupid obviousness, and it's probably not one said with fondness, and yet I can't help but detect a whiff of profundity there. Though perhaps it would be clearer for me to say, "If there is only a rock, then there is only a rock." To be more precise, it doesn't matter if there is anything else with which to use as a frame of reference; if a rock exists, then dammit, it exists. (I ultimately changed the rock to a tomato, because I found that funnier, which is how the poor tomato became involved.)

And to say that "if reality is finite you could describe it in terms of a tomato" is simply to say that if one could *somehow* observe all of a finite reality at once, and furthermore, had an intelligence far greater than and could calculate infinitely faster than the greatest theoretical supercomputer, then it should theoretically be possible for this intelligence to assign a value to every property of everything in existence as it relates to everything else in existence. So while a tomato has the obvious properties of being "red" and "soft" and "vegetable", so must I have some kind of less obvious value for these properties, even if my value is zero or even negative (or even imaginary? I've never really understood the concept of imaginary numbers, although I've never really put much effort into understanding them.)

So I figured that if you must pick a "ground zero," so to speak, with which to find common denominators for the entirety of reality, you might as well start with a rock, or better yet, a tomato-why-not.

Of course, above I only mentioned physical properties. I first starting thinking of all of this by assuming that if there was a supremely omnipotent God, one who could observe all of a finite reality at the same time, he could potentially see everything as one huge, unimaginably complicated and constantly changing mathematical equation. My "redness," my "softness," but also, since we're talking about a supremely omnipotent God who observes *all* of reality, we have to include "my love for my cat," "my anger at being splashed by that puddle," "this thought I'm thinking right now;" emotions, thoughts, and all sorts of intangible things that I can't observe but that are a part of reality as we know it nonetheless (and as such is a part of what a supremely omnipotent God should be able to observe and therefore assign a value to.) Because it seems to me that even thoughts, emotions, etc. exist in some sense, even if I can't prove that, and even if they're neither observable nor measurable. I mean, I'm fairly certain I'm thinking right now...? (Or maybe I just think that I'm thinking.... errrr.... *head assplode*)

Of course, if I understand what I'm saying correctly here (admittedley doubtful), than this approach would necessitate finding what would probably be close to an infinite amount of common denominators (properties?) between observable (what we ourselves can observe) and unobservable reality. (To be clear, I'm saying that "observable reality"+"unobservable reality" = "reality," the totality of which a supremely omnipotent God could observe). Which isn't really possible, but theoretically, as I stated before, if there was such a thing as infinite wisdom combined with supreme perception, it seems it could be possible if you realize that most of the values assigned to the properties of intangibles would have a negative or zero (or possibly even imaginary?) value when applied to tangibles, and vice versa.... but those values would still, in some sense, exist. Err, maybe.

It got really out of hand when I considered that consciousness, and by extension, free will, as part of the totality of reality, would have to have a place in this huge mythical equation describing all of reality. I then decided that it would be ridiculously impossible for me (or possibly anyone who's not completely insane) to write "free will" as an equation. But then I got into reading about "choice functions" (or whatever they may be called, something about an infinite number of bins and deciding which bin to place a package in or something to that effect) and that's where I gave up because I was in danger of losing my mind (and didn't really understand what I was reading anyway, since the more complex "formulas" in math are basically just sentences and truths written in a language I don't know how to read.)

It is also interesting (to me, anyway) to note that *if* some kind of observation is in fact necessary to exist, and *if* in fact a rock that exists does so with or without an outside observer and by implication must be observing itself in order to exist, then I have assumed a certain amount of consciousness on the part of the rock. Err, excuse me... tomato. (I called it "awareness," rather than consciousness, to make it sound less stupid, but really, it might as well be the same thing.) But this is when I started thinking that if free will exists, than everything within reality must have some kind of value for free will, including the tomato and all of it's smallest particles - and also, in order to exist, particles must have some kind of fundamental awareness of themselves even if it's so miniscule that we could never hope to comprehend or measure it. (I also became fond of thinking that the reason the smallest observable particles sometimes seem to flicker in and out of existence (I read this somewhere, I believe) is simply because they aren't aware enough of themselves or their environment to understand the difference between existing and not existing. Sometimes, they cease to exist because they choose to, but more importantly, because they don't really know any better.)

I just realized that I'm basically expounding on the classic "I think therefore I am," although really I'm saying that "A particle thinks, and therefore it is... except that sometimes it doesn't, and therefore, at those moments, it isn't." This is probably all nonsense, but dammit, it's poetic nonsense. But I made the leap that if true, then choice, or free will, these intangible things, may *be* the Higgs Boson (that's the thing that gives matter its form, right? That thing we can't seem to find? If I recall correctly and if I'm not being too simplistic.) What I mean to say is that maybe we can't find it because we're looking for an intangible, a choice: matter is able to take a certain form simply because its smallest particles, in some rudimentary sense, develop enough of an awareness to continue existing, and then, in some abstract sense, choose to take a particular form.

Anyway, sorry, I'm rambling and off-topic here, and I better stop with this because I'm beginning to confuse myself, which is probably a sign that I'm delusional. Maybe this all belongs in a different thread... perhaps "humour?" I really didn't intend to go into all of that, but once I started I couldn't help but try to explain myself. I mean, it's not often I get the chance to show to physics (or logic) buffs just how little I actually understand about their respective fields. I'm quite sure that most of the above is nonsensical and makes conclusive leaps that it shouldn't, and I probably contradict myself without realizing it, but I'm not convinced that this is because I'm wrong, it may just be that I'm incapable of proving or disproving anything because I don't know enough to detail logical steps from premise to conclusion. But I do realize that both could be true; I may be in danger of being insane, and completely unschooled, but I'm fairly certain I'm not stupid.

Against my better judgement, and at the risk of embarassing myself, I'm about to hit "submit reply." Just do me the favor of laughing with me, and not at me. (And sorry for the novel; that's just what I do. I've always been under the impression that the more ways I can repeat myself, the clearer I will be. It's a condition.)

[Sage Lee]

Deuterium, I saw that you posted while writing the above, and have not yet had time to do more than skim. I'm going to have to read it a few times before responding to any of it, because although I think I get the gist of most of what you are trying to explain, I still got a little lost amongst the gobbledygook. (You only thought you knew the meaning of that word till I showed up...)

But if you'll excuse me, the poker staking forum I belong to is having an interesting discussion about why or why not God exists, what he would be like, and the overall nature of good and evil. I'm currently trying to explain why I think it might not be possible for Heaven to be "better" than our current condition. (My attentions tend to wander, so, like the butterfly, I must float...)

[Sage Lee]

Also, while waiting for a response in that other forum (sometimes I wonder if any of those guys actually ever play poker) I couldn't help but click a "related thread" link that I found below, and I found this guy talking about his theory of infinite-infinite:

http://www.physicsforums.com/showthread.php?t=65278

This is basically what I feel is "disproven" by the simple fact that all of reality isn't happening everywhere and at the same time, over and over again. There HAS to be some kind of limit to reality, or it would be an inconceivable and chaotic blur of all possibilities happening at once, running into each other (occupying the same space and time. And everything else.)

In my humble opinion, anyway. I can't prove that. Sigh.

[Chalnoth]

Okay... honestly, I kind of rolled my eyes (at first) when I saw that you just rehashed the same example as had been stated previously, using a finite list of numbers (except this time you said "1 through 5" instead of "only even numbers". Because this is where I suffer a disconnect: you're saying, it's infinite!!!!..... eeeeexcept it stops at 5. To my way of thinking, saying it stops at 5 is already cheating, because if it stops at 1 on one end and 5 on the other end, it's not really infinite, is it? It's limited and therefore in a sense *finite* in that it can only use five numbers.
It doesn't stop, though. It keeps repeating over and over again, endlessly.

In a real-world scenario, this would be like there being an observable universe somewhere far away that is absolutely identical to our own. If the universe is infinite, in fact, we know this must be the case, because due to quantum mechanics there are only a finite number of possible configurations. So if it is infinite in space, then the real universe would actually behave very much like the repeating number line, except that the repetition would be more chaotic than orderly.

If we were to use numbers to represent an infinite reality, I would've thought that we must by definition have no limits on the numbers we choose to use, if we're talking about infinite. Like, an infinite representation using only numbers would by definition have to include all numbers, positive and negative, odd and even, real and imaginary, all integers and complex numbers and everything in between; it would go on forever in all directions, with no finite "bookends," so to speak (1 and 5). And there would be an infinite amount of 1's and 2's and 6's and 10's and an infinite amount of each negative number and imaginary number and so on... or else it wouldn't be an accurate representation of "infinite."
From quantum mechanics we find that the total number of possible configurations of a given region of the universe is finite. It's a very large number, but a finite one nonetheless.

Yes, yes, of course... believe me, I do understand this. But there is still a part of me that wants to say "just because my puny mind can't comprehend the existence of something that has only three sides while still *somehow* having four sides doesn't mean that it's not possible."
Well, no, because in this case a triangle is an abstract mathematical construct. It isn't a real object. Because it is an abstract mathematical construct, with a very specific definition, we can say absolutely that it doesn't have four sides.

And yet the bolded has always bothered me, because yes it is easier but not necessarily correct.
No, that's not the right way to look at things. Our choice of logic is more or less arbitrary. One choice of logic is no more or less correct than another. But one choice may be more useful than another under certain situations.

[Sage Lee]

Sage, you may be mixing up two concepts...that of a Line, and that of a Line Segment. By it's very nature, a Line (in the strict geometric sense) is infinite in length. A Line Segment is bounded, and of finite length.

I don't think I mixed these two up, rather, I was arguing that an infinite line running east to west is similarly bounded, albeit in a different fashion, in that it can't ever bend or travel north or south, up or down.

But suprisingly, I (think that I) actually get most of what you told me in this post, on my second read through. You're basically saying (I probably think about this weird, but I think the conclusions are the same) that a line segment and a line, though one might be smaller than the other, are both infinite in the sense that you can theoretically zoom in enough (for lack of a better way to say that) on any given section of pretty much anything, and plot an infinite number of points.

And in this sense, it also seems like you just told me that infinity is contained within finite things, you just have to be capable of going smaller and smaller. Yikes. You're crazy, man. I like you, but you're crazy. (No, just kidding.)


For those unfamiliar with Set Theory, it comes as a real shock to learn that there are EXACTLY the same number of points on the line interval from [0,1] as there are on an interval twice as long [0,2].

I guess I could say I was familiar with Set Theory, since it drives me batshit crazy. My stumbling upon Set Theory is actually directly responsible for my attempt at Masturbational Existentiality; I still remember the first thought that I had when I read about Set Theory, it was something along the lines of, "Holy ****, you can talk about anything as math, even abstract or intangible things!" (This may not really be true, but at the time it got me thinking about a supremely omnipotent observer, and what he might or might not be able to observe, and how to quantify all of what he could possibly observe (including intangibles.) It was my discovery of Set Theory that started that whole train wreck line of thinking.

So, to clarify, is it possible to talk about sets containing abstract things, like "the set of all thoughts about hot dogs," or can you only have a set containing objects? Some of the things I said when talking about M.E. a few posts ago are probably even more ridiculous than I realized, as I thought at the time that such intangibles could already be quantified using Set Theory... but now I'm realizing this might not be true. Mehhh, but I so want it to be true!


Perhaps the single biggest surprise, when first learning transfinite Set theory, is that not all infinite Sets are equal.

This was actually not that hard for me to stomach, as it seems to make perfect sense once explained correctly.

For me, the biggest surprise was that an empty set has a cardinality of 1. (Did I say this right?) This just pissed me off, and got me reading about vacuous truth, and it wasn't long before I threw my hands up in exasperation and stopped trying to understand why.

But because of my frustration, I didn't like the joke "in a set of zero mathematicians, any one of them can do it [change a light bulb]." I actually remarked, to no one in particular, that "in a set of zero mathematicians, three of them are actually tomatoes." I liked this better because, "Hey, if we're being ridiculous, let's just let it all hang out and be ridiculous." What can I say, I was annoyed and was on that previously described tomato kick at the time.

But whatever, I accept on faith alone that an empty set is actually "one," because Wikipedia told me so... but I don't have to like it.

(You have to keep in mind that I and my unschooled mind tried to take in a LOT of very complex information all at once, pretty much on a whim (damn this insatiable curiosity I have to understand,) and for this reason, it's very hard for me to retain much of it. Also because it's not like I ever put any of it into practice, I just thought about it for a while. This was all about five years ago; I don't really remember exactly why I had such a problem with the empty set, or why I said those things I said, I just remember saying them.)

But all in all, I really, really like Set Theory, because as I said, with it, it seems possible to describe just about anything at all using math.

Okay, I just have to share the other joke I came up with when I first read about Set Theory. Alright, ready?

N > Stephen Hawking

I find this funny, but only because I know what N is. In all honesty, I should probably just leave it at that, because if I tell you what N is you'll just think I'm an *******. And besides, nothing is as funny if you have to explain it.

But *sigh* I started it, so I'll finish it: N is the set of all things that can change a light bulb.

Now, to be clear, I don't mean this in any spiteful kind of way. Obviously I can't relate to being in a wheelchair, and I certainly don't understand how it might feel to have that poked fun of, but I really don't mean to be malicious with that joke. I don't intend to slight the man himself in any way; in fact, I'm quite convinced that he can probably shoot laser beams out of his eyes and crumble my very existence with a single, profound thought. Hell, who needs to change light bulbs when you can power them forever with your mind? Rather, I'm poking fun of the absurdity of such a brilliant and existence-crumbling-mind being (probably) unable to accomplish such a simple task (without assistance), one that much simpler folk like myself take for granted.

Forgive me, but I pretty much find everything funny given the right delivery or moment. I'd like to think that if Hawking heard that joke, he'd be wise enough to be able to take it in the spirit it's meant, and to maybe even also find it funny. I don't know, does anyone else find my joke funny, or should I just keep things like that to myself?

Regardless, I still think that would make a great T-shirt (just the joke, without the explanation of what N is.) Visually, to non-math people, it reads "N is greater than Stephen Hawking" (rather than N contains Stephen Hawking) and at it's core is saying, in a roundabout way, that "a light bulb is greater than Stephen Hawking." Frankly, I just find the thought of ANYTHING being greater than Stephen Hawking to be kind of funny, who cares what N actually is?! I would wear the **** out of that shirt, and if anyone asked me what it meant, I'd probably just smile and shake my head. (I'm also aware that "N" in math might already mean something specific, but if you can choose whatever letter you want to designate some set you just pondered, then I choose N, as it's better visually for me than A or B or X or Y or Z. Don't ask me why; I'm particular about these things.)

In my final defense, I'll just point out that I don't find this hilarious or anything, it just makes me smile.


In order to understand this, you need to understand the formal, logical distinction between what is a "necessary" condition, and what is a "necessary AND sufficient" condition. They are not the same. I guess the best way to explain is through an analogy and example.
To say that X is a necessary condition for Y is to say that it is impossible to have Y without X. In other words, the absence of X guarantees the absence of Y.
Example: Having four sides is a Necessary condition for being a Square.
Notice, however, it is not a Sufficient condition. For example, a Rectangle has four sides, as does a Rhombus, but they are not necessarily Squares. A Rectangle has four equal angles, but may not have four equal sides. Conversely, a Rhombus has four equal sides, but may not have four equal angles.
Compare/contrast the above example to the following:
A quadrilateral with four equal sides and four equal angles is a both Necessary and Sufficient condition for being a Square.
-Or- another way of phrasing this: A quadrilateral that is BOTH a Rectangle AND a Rhobus is a Necessary and Sufficient Condition for being a Square.

Out of curiosity, is it correct to capitalize all those words when using math-speak? It never would've occured to me that it's proper to capitalize Rectangle, but since you took the time to do it in several instances, now I'm thinking it's probably the norm. I find that interesting. As you may have realized, I write a lot, but I don't recall ever having cause to write the word Rectangle.


Now, getting back to your question as to how an Infinite Universe isn't a "Sufficient" condition for "Everything existing somewhere"...
It is a Necessary condition that the Universe be Infinite in order for there to exist the possibilty that "everything exists somewhere". This is obviously trivially true, because if it were not infinite, then it would be finite, and a finite Universe cannot be a Necessary condition for everything existing somewhere. So, as a minimum, it is a Necessary condition that the Universe be Infinite in order for this possibility to exist. However, that is not a Sufficient condition. As discussed in earlier posts in this thread, the Universe may be "countably" infinite...that is to say, having the same size (Cardinality) as the countably Infinite Set of Natural Numbers ( |N| ). However, the Set of all Even numbers is just as big (i.e. the same size) as the Set of all Natural Numbers, yet the former Set is missing an infinite amount of numbers...that is, the Odd numbers. So, these two sets have exactly the same NUMBER of elements (members), but these two sets are not "identical", and only one of these sets "exhaust" all the Natural numbers, whereas the other set does not.

I think infinity just doesn't mean what I thought it did at the start of all this. It's still kind of bothersome that something can be infinite and yet be missing an infinite amount of things, but I think I get it now.


With that said, I am not exactly certain what would be both a Necessary and Sufficient condition for an infinite Universe to ensure that "everything exists somewhere". From a purely mathematical perspective, I might argue that the Universe would need to have the Cardinality of the Continuum

Wow, that sounds really cool. If I had to name a band, or an album or something, right now, I'd name it that. It sounds so damn epic.


(= the Set of Real numbers). However, one could equally argue that that, in and of itself, may not even be a Sufficient condition. The tiny interval [0,1] on the Real number line is everywhere Dense and Continuous, and this segment contains an equal number of points as in the entire Real Number line. In fact, it contains in equal number of points as on a plane. Moreover, it contains just as many points as on any finite n-dimensional space. Nevertheless, despite the equipollence of the interval [0,1] with the entire Real Number line, it is not "exhaustive". It doesn't contain the number "2", or "pi", or "e", or for that matter any Real number greater then one or less then zero.

I don't *quite* get what you mean by "dense" here, although I think you're just reiterating what you've already explained in a slightly different way.


All this gobbledygook ultimately comes down to the conclusion that, even though the Universe may be infinite, it does not necessarily follow that "everything exists somewhere".

Congratulations, to both you and Chalnoth. I now completely agree with that statement. Gold star for youse guys. Although I'm thinking, as I said before, that I never really disagreed, I just didn't understand what infinity actually meant (I thought it literally meant "exhaustive.")

[Sage Lee]

It doesn't stop, though. It keeps repeating over and over again, endlessly.

Yeah, but it's fiiiiniiiite! <stamping foot and holding breath>

In a real-world scenario, this would be like there being an observable universe somewhere far away that is absolutely identical to our own. If the universe is infinite, in fact, we know this must be the case, because due to quantum mechanics there are only a finite number of possible configurations. So if it is infinite in space, then the real universe would actually behave very much like the repeating number line, except that the repetition would be more chaotic than orderly.

Wow, I didn't even consider this implication. How very interesting.

From quantum mechanics we find that the total number of possible configurations of a given region of the universe is finite. It's a very large number, but a finite one nonetheless.

We are on the same page...


Well, no, because in this case a triangle is an abstract mathematical construct. It isn't a real object. Because it is an abstract mathematical construct, with a very specific definition, we can say absolutely that it doesn't have four sides.

This seems like a funny thing to say. If we were talking about a real object, wouldn't it have an even more (or at least an equally) specific definiton, and couldn't we also say absolutely that it doesn't have four sides? I mean, having something to look at and touch and feel seems like it would be more definitive than just thinking about an abstract concept, so I find it weird that you started that sentence with "because it's not real" Of course, I don't deal much in math, so that's probably why that seems that way to me.

But I get it, I can't really argue with anything you've said on this subject, even if I like to try and play Devil's Advocate.

What you're saying is, you don't believe in magic. (No, don't respond to that, I'm just playing now, and besides, dead horses start to smell after a while, so I'll just sweep this one under the rug and move on...)

No, that's not the right way to look at things. Our choice of logic is more or less arbitrary. One choice of logic is no more or less correct than another. But one choice may be more useful than another under certain situations.

This, though I don't really get. I should probably brush up on my logic. By which I mean to say I need to go back to 101. I never got beyond "if," "then," and the occasional "but," plus I skipped that class all the time and ultimately dropped out because I preferred to smoke weed and play hackisack before lunch. (Oh, who am I kidding, I skipped all my classes, no matter the time of day. In my defense, it was some pretty good weed, and I was a ****in hackisack god. Fortunately, I squeaked by in a few classes because, believe it or not, I'm quite charming in person, and my teachers have mostly seemed to like me. I've been told I have charisma, whatever that means. If it was a girl who said it, it would probably mean "I'm ugly," but fortunately, it wasn't.) Aaaaaanyway, I thought logic was logic, and as you stated before, is consistent. What are these choices of which you speak, and how is that choice arbitrary? Do you care to provide any simple examples of what you just said?

No biggie, if not; after reading that other thread I linked in an earlier post, the one about infinite-infiinte, I couldn't help but wonder how often some new guy comes in here and just up and barfs all over the forum, leaving you guys to clean up the mess, and I appreciate the patience you must have when dealing with people like me. So I understand if not, and I can probably find some good places to learn on the internet, but it's always nice to be able to ask questions and further refine one's knowledge.

[Chalnoth]

This seems like a funny thing to say. If we were talking about a real object, wouldn't it have an even more (or at least an equally) specific definiton, and couldn't we also say absolutely that it doesn't have four sides?
Any time we apply mathematics to reality, we have to consider that we don't actually know for certain whether or not the mathematics applies.

In this situation, for instance, there's no such thing as a triangle in reality. You can draw something that looks like a triangle on a piece of paper with a pencil, for instance, but what it really is is a bunch of graphite and rubber atoms spread across the surface of the paper. It simply isn't possible to make atoms form a line segment, because the atoms are of finite size.

Because of this, it is very possible to draw something that looks like a triangle on paper, but doesn't actually have all of its properties.

This, though I don't really get. I should probably brush up on my logic. By which I mean to say I need to go back to 101. I never got beyond "if," "then," and the occasional "but," plus I skipped that class all the time and ultimately dropped out because I preferred to smoke weed and play hackisack before lunch. (Oh, who am I kidding, I skipped all my classes, no matter the time of day. In my defense, it was some pretty good weed, and I was a ****in hackisack god. Fortunately, I squeaked by in a few classes because, believe it or not, I'm quite charming in person, and my teachers have mostly seemed to like me. I've been told I have charisma, whatever that means. If it was a girl who said it, it would probably mean "I'm ugly," but fortunately, it wasn't.) Aaaaaanyway, I thought logic was logic, and as you stated before, is consistent. What are these choices of which you speak, and how is that choice arbitrary? Do you care to provide any simple examples of what you just said?
What is arbitrary about logic is what sorts of statements we allow into the logic. Once we have defined the allowable statements, everything else is exact. So when applying logic to the real world, we need only make sure that we restrict ourselves to the allowable statements in the logic.

For example, in classical, first-order logic, the only allowable statement has the property that it is either true or false. Once you have that set up, the rest of the rules necessarily come about due to consistency: since the only allowable statements are true or false, a set of logic rules that leads to contradictory results is invalid.

In practice, this is how logical fallacies are discovered: we find a counter-example to the argument.

Finally, let me state that logic is just a way of thinking about the world. With logic, we take a series of propositions, and determine what can be drawn from those propositions. For example, if I take the propositions:
All boys have brown hair.
Bob is a boy.

...then I can infer that Bob has brown hair. Pure logic can never actually say whether the propositions or the conclusion(s) of a logical argument are true. But what it can do is link different propositions and conclusions together. In practice, we have to go out and look at the world to see whether or not our propositions or conclusions are true. For example, in the above case, if I look at Bob and find that he doesn't have brown hair, I now know that one of the two propositions must be wrong (either Bob is not a boy, or at least some boys don't have brown hair). The only uncertainty here is in my observation of Bob's hair color: I am equally as sure that one of the two propositions is wrong as I am sure that Bob doesn't have brown hair. There is no uncertainty in the logical deduction.

[Sage Lee]

Any time we apply mathematics to reality, we have to consider that we don't actually know for certain whether or not the mathematics applies.

In this situation, for instance, there's no such thing as a triangle in reality. You can draw something that looks like a triangle on a piece of paper with a pencil, for instance, but what it really is is a bunch of graphite and rubber atoms spread across the surface of the paper. It simply isn't possible to make atoms form a line segment, because the atoms are of finite size.

Because of this, it is very possible to draw something that looks like a triangle on paper, but doesn't actually have all of its properties.

Okay, makes sense

What is arbitrary about logic is what sorts of statements we allow into the logic. Once we have defined the allowable statements, everything else is exact. So when applying logic to the real world, we need only make sure that we restrict ourselves to the allowable statements in the logic.

For example, in classical, first-order logic, the only allowable statement has the property that it is either true or false. Once you have that set up, the rest of the rules necessarily come about due to consistency: since the only allowable statements are true or false, a set of logic rules that leads to contradictory results is invalid.

In practice, this is how logical fallacies are discovered: we find a counter-example to the argument.

Finally, let me state that logic is just a way of thinking about the world. With logic, we take a series of propositions, and determine what can be drawn from those propositions. For example, if I take the propositions:
All boys have brown hair.
Bob is a boy.

...then I can infer that Bob has brown hair. Pure logic can never actually say whether the propositions or the conclusion(s) of a logical argument are true. But what it can do is link different propositions and conclusions together. In practice, we have to go out and look at the world to see whether or not our propositions or conclusions are true. For example, in the above case, if I look at Bob and find that he doesn't have brown hair, I now know that one of the two propositions must be wrong (either Bob is not a boy, or at least some boys don't have brown hair). The only uncertainty here is in my observation of Bob's hair color: I am equally as sure that one of the two propositions is wrong as I am sure that Bob doesn't have brown hair. There is no uncertainty in the logical deduction.

Thanks, good explanation.

[Sage Lee]

I just stumbled across this thread:

http://www.physicsforums.com/showthread.php?t=59347&page=2

Where in post 19 someone talks about what I was trying to talk about but in a much more intelligent fashion. But I believe he points out that "it's only a consistent way of talking about reality because it misrepresents it" or something to that effect. Which still makes this all kind of pointless. Plus, that was 5 years ago, so maybe the works he's referencing have already been laughed off the table.

[Chalnoth]

I just stumbled across this thread:

http://www.physicsforums.com/showthread.php?t=59347&page=2

Where in post 19 someone talks about what I was trying to talk about but in a much more intelligent fashion. But I believe he points out that "it's only a consistent way of talking about reality because it misrepresents it" or something to that effect. Which still makes this all kind of pointless. Plus, that was 5 years ago, so maybe the works he's referencing have already been laughed off the table.
Well, while strictly correct in terms of mathematical/logical proof, what he wrote is very misleading. While we can never prove whether idealism or materialism is correct, we can obtain evidence that favors one or the other possibility. Materialism states that there exists a self-consistent reality external to ourselves which we perceive, however imperfectly. Such a reality, because it must be self-consistent, will contain patterns that allow us to make use of inference. Every time such inference is successful, we gain confidence that materialism is accurate. The success of modern science, then, provides a vast array of evidence in favor of materialism.

Idealism, on the other hand, which asserts that there is no way of knowing whether or not our putative observations are imaginary, possesses no such constraints. Imaginary worlds are not limited in any sense of the word, so that if we think we see some patterns, and make some predictions based upon those patterns, we may expect that sometimes those predictions will succeed, but usually they will fail, and if we wait long enough, those predictions will always fail, if idealism is accurate.

So when we have a scientific theory, such as Newtonian mechanics, that has stood the test of time, continually and repeatedly providing accurate answers to the same sorts of experiments, we have extreme confidence that idealism cannot be true.

We can never prove it, of course. This is the basic problem of inference. But the more our inference works, the more confident we are that it's a good way of doing things.

[GODISMYSHADOW]

Well, consider this by way of analogy.

The set of all even numbers is infinite. I can go on counting even numbers for ever and ever and never reach an end.

But clearly the set of all even numbers does not include all possible numbers. It doesn't include, for instance, the number pi.

So even if the universe is infinite (we don't know whether or not it is), then that doesn't necessarily mean that all possibilities are realized.

However, there may be other reasons to believe that all possibilities are realized, mainly stemming from quantum mechanics, where we find, for instance, that if there is the possibility of matter inhabiting a region of space, then particles of that sort of matter will necessarily pop in and out of the vacuum. Another way of saying this is that in quantum mechanics, there mere possibility of existence forces existence. So it is not unreasonable to suspect that perhaps all possibilities must actually be realized.

This doesn't mean that anything and everything we can imagine occurs, of course. We can imagine quite a lot of impossible things, as you mention above. But we can also imagine a great many things that are not obviously impossible, and yet may turn out to be upon deeper inspection.

Would the Universe be "the set of all things right now at this moment"? That can't be right, because Einstein showed there is no "absolute time" and hence no "absolute now". Could that mean there's really no Universe?

[marcus]

Would the Universe be "the set of all things right now at this moment"? That can't be right, because Einstein showed there is no "absolute time" and hence no "absolute now". Could that mean there's really no Universe?

"absolute" just means something that all observers agree on---it does not depend on the observer and his motion relative to other observers.

Just because you can have disagreement between observers (i.e no absolute time) doesn't mean the U doesn't exist.

However the phrase "right now at this moment" (that you used) does depend on what observers you are talking about----it takes some discussion.

The universe can exist just fine and yet different sets of observers can have different ideas about how to slice it into Present Moments.
==============================

I'll throw in some extra detail just in case anyone is curious to follow this further.

In cosmology we have a special set of observers!
A preferred perspective on the universe. So a preferred idea of simultaneity, and a time sometimes called "universe time" or "Friedmann model" time.

This set consists of all observers who are at rest relative to the ancient light.
The glow of ancient matter, from when the universe was just uniformly filled with hot gas. This glow is now the microwave background or "CMB".
An observer is at rest relative CMB if he perceives no big doppler hotspot ahead of him or coldspot behind. If he measures the temp approximately uniform in all directions.

We could have a network of observers all over the universe, all at rest relative CMB, and they could all synchronize their clocks! They could all agree on a slicing of events into synchronous slices. And they could all agree on the age of the universe.

Observers moving relative CMB would not agree, unless they compensated for their motion and took the viewpoint of a stationary observer.

And in fact that is what we do. We know the earth's speed and direction relative CMB and we CORRECT observational data for that. We adjust so we can have data that is from the standpoint of a stationary observer. It is a very tiny correction because we are almost stationary, so in most situations you can neglect it.

But in a certain sense there is, in cosmology, a practical idea of an "absolute" time, or at least pragmatically preferred time, that the standard Friedmann equation model runs on, and corresponds to stationary observers time.

General Relativity allows this. The point is we have a kind of landmark. The glow from the ancient matter. Matter is what makes the difference.

[A. Neumaier]

In a real-world scenario, this would be like there being an observable universe somewhere far away that is absolutely identical to our own. If the universe is infinite, in fact, we know this must be the case, because due to quantum mechanics there are only a finite number of possible configurations. So if it is infinite in space, then the real universe would actually behave very much like the repeating number line, except that the repetition would be more chaotic than orderly.

From quantum mechanics we find that the total number of possible configurations of a given region of the universe is finite. It's a very large number, but a finite one nonetheless.

How did you arrive at this view?

How do you define a ''possible configurations of a given region of the universe'' in such a way that you can count their number?

[Chalnoth]

How did you arrive at this view?
Well, I thought I explained it sufficiently. Infinite space + finite configurations = repeating universe.

How do you define a ''possible configurations of a given region of the universe'' in such a way that you can count their number?
Well, there are a few ways to go about it. From one direction, we can approach the issue from the side of entropy, as entropy is proportional to the logarithm of the number of states that can replicate the macroscopic properties of the system (though this has the problem that we don't know how to calculate the entropy for every macroscopic configuration). From the other direction, we can approach the issue from quantum mechanics and just count the number of states that are available. This has the problem that we don't know the behavior at very high energies.

But in any event, the result, if we knew how to calculate it, would have to be finite in any case, because the entropy is finite and an infinite result for the quantum mechanical calculations would lead to nonsense in calculating simple things like reaction cross sections.

[A. Neumaier]

Well, I thought I explained it sufficiently. Infinite space + finite configurations = repeating universe.

I meant, why do you think that there are only finitely many configurations in an infinite universe?

From one direction, we can approach the issue from the side of entropy, as entropy is proportional to the logarithm of the number of states

the entropy is finite and an infinite result for the quantum mechanical calculations would lead to nonsense in calculating simple things like reaction cross sections.[/QUOTE]

A finite entropy density in an infinite universe may well lead to an infinite total entropy.

[Chalnoth]

A finite entropy density in an infinite universe may well lead to an infinite total entropy.
Yes, but we're not talking about total entropy, but rather the entropy of an observable region. And as long as the entropy density is finite, the entropy of an observable region of any given size will also be finite.

[A. Neumaier]

Yes, but we're not talking about total entropy, but rather the entropy of an observable region. And as long as the entropy density is finite, the entropy of an observable region of any given size will also be finite.

But the states in different observable regions may be different! Entropy doesn't tell you anything about that. (Otherwise, bu reducing the observable regions sufficiently, you could make the total number of distinct states as small as you like.

Moreover, there are vastly more states than the energy eigenstates counted by the entropy. Most observable systems are not in an energy eigenstate but in a complex superposition of these - and there are infinitely many possibilities for these, already for a single qubit.

[GODISMYSHADOW]

"absolute" just means something that all observers agree on---it does not depend on the observer and his motion relative to other observers.

Just because you can have disagreement between observers (i.e no absolute time) doesn't mean the U doesn't exist.

However the phrase "right now at this moment" (that you used) does depend on what observers you are talking about----it takes some discussion.

The universe can exist just fine and yet different sets of observers can have different ideas about how to slice it into Present Moments.
==============================

I'll throw in some extra detail just in case anyone is curious to follow this further.

In cosmology we have a special set of observers!
A preferred perspective on the universe. So a preferred idea of simultaneity, and a time sometimes called "universe time" or "Friedmann model" time.

This set consists of all observers who are at rest relative to the ancient light.
The glow of ancient matter, from when the universe was just uniformly filled with hot gas. This glow is now the microwave background or "CMB".
An observer is at rest relative CMB if he perceives no big doppler hotspot ahead of him or coldspot behind. If he measures the temp approximately uniform in all directions.

We could have a network of observers all over the universe, all at rest relative CMB, and they could all synchronize their clocks! They could all agree on a slicing of events into synchronous slices. And they could all agree on the age of the universe.

Observers moving relative CMB would not agree, unless they compensated for their motion and took the viewpoint of a stationary observer.

And in fact that is what we do. We know the earth's speed and direction relative CMB and we CORRECT observational data for that. We adjust so we can have data that is from the standpoint of a stationary observer. It is a very tiny correction because we are almost stationary, so in most situations you can neglect it.

But in a certain sense there is, in cosmology, a practical idea of an "absolute" time, or at least pragmatically preferred time, that the standard Friedmann equation model runs on, and corresponds to stationary observers time.

General Relativity allows this. The point is we have a kind of landmark. The glow from the ancient matter. Matter is what makes the difference.

You're saying this "CMB" is used as a reference frame in your cosmology.
I'm going to have to study this stuff to gain a better understanding.

An event in the forbidden zone has no causal effect on my here-now because it's
outside the light cone. (That's absolute elsewhere on the Minkowski diagram.)
It's important to consider for astronomical distances. However, an event in the
forbidden zone may have a causal effect on some event happening in my future.
That being the case, if the universe is a set of events in the forbidden zone, then
the universe can't be more real than events in my future. That invites the question,
"Does the future exist?" Some say we can change our destiny if we try. Others say
the future is already there, it's irrevocable and cannot be changed. I wonder.

[A. Neumaier]

"Does the future exist?" Some say we can change our destiny if we try. Others say
the future is already there, it's irrevocable and cannot be changed. I wonder.

This is undecidable.

Suppose you'd record every detail about the history of the universe, wait till it has died, and then replay it in a perfect simulation (where of course, everything is already there). The physical laws would be exactly the same - without the slightest detectable difference.

[TrickyDicky]

....there is, in cosmology, a practical idea of an "absolute" time, or at least pragmatically preferred time, that the standard Friedmann equation model runs on, and corresponds to stationary observers time.
General Relativity allows this. The point is we have a kind of landmark. The glow from the ancient matter. Matter is what makes the difference.

Nicely put Marcus. I'd like to add to your insightful phrase in my bold that maybe sometimes we get hung upon abstractions about Spacetime, but it's good to remember ourselves once in a while that spacetime is just a geometrical abstraction to describe the relations within matter in its broad meaning of mass-energy continuum. In this sense matter is all there is and surely what makes the difference.

About the stationary observers, they illustrate the way the GR equations were designed in a general covariant way to have 6 independent differential equations with 6 unknown quantities and another 4 unknown quantities that are arbitrarily fixed with the choice of coordinates.
This condition allows us to stablish the rest frame or stationary observers as we set the coordinate space and the coordinate time for a particular metric, and therefore we can determine a rest state wrt these coordinates so in this sense the fundamental observers appear not only in the "Friedmann model" but in any metric we might build from the GR equations.

In our cosmological model this rest frame is embodied by the CMB like you say, we measure our motion with respect to this radiation that fills the vacuum thru the universe.

This is for a very practical reason, the CMB are photons and we are able to detect them, quite easily (from 1965 at least), we could say the CMB is the "visible" part of the energy density of the vacuum, which is indirectly observe or "felt" as dark energy (and also as dark matter according to some models with inhomogeneities such as those of T. Buchert et al., but these models are not mainstream).

[Chalnoth]

But the states in different observable regions may be different!
Yes, very true. So to do this properly, you'd have to integrate over all macrostates. That result, also, will have to be finite.

Moreover, there are vastly more states than the energy eigenstates counted by the entropy. Most observable systems are not in an energy eigenstate but in a complex superposition of these - and there are infinitely many possibilities for these, already for a single qubit.
The specific superposition of states is just a representational issue and thus cannot be a physical effect. That is to say, a particle that is in an eigenstate of energy is in a superposition of states in position. So you can recast any particle that we "see" as being in a superposition of states as being in a particular eigenstate by constructing your operator appropriately.

[A. Neumaier]

Yes, very true. So to do this properly, you'd have to integrate over all macrostates. That result, also, will have to be finite.

Nothing in quantum mechanics allows you to deduce this!


The specific superposition of states is just a representational issue and thus cannot be a physical effect. That is to say, a particle that is in an eigenstate of energy is in a superposition of states in position. So you can recast any particle that we "see" as being in a superposition of states as being in a particular eigenstate by constructing your operator appropriately.

But entropy only counts the eigenstates of the energy. On the other hand, most states in nature are not eigenstates (only stationary states are). Thus the vast majority of observable states is not counted by entropy.

[Chalnoth]

Nothing in quantum mechanics allows you to deduce this!
This stems from the exact same arguments as in quantum field theory: there has to be some high-energy cutoff.

But entropy only counts the eigenstates of the energy. On the other hand, most states in nature are not eigenstates (only stationary states are). Thus the vast majority of observable states is not counted by entropy.
Now you're mixing different descriptions of the same system. But it isn't true in any event. The computation of entropy has to count the full set of microstates, which for real particles also includes things like spin and angular momentum, as well as energy.

[Deuterium2H]

Coming full circle, and getting back to the original question/post...I think that my arguments using mathematically-based Set Theory, and Chalnoth's physics-based arguments (Thermodynamics, Statistical and Quantum Mechanics) have both converged on an answer that is rather non-intuitive. Certainly, it goes against popular "opinion". But if mathematics can teach us anything, it is that transfinite Set Theory is itself counter intuitive. This just so happens to be very much the case, as well, with Quantum Theory.

The answer to the the original post is quite simply this...

Given an infinite Universe, it is does NOT necessarily follow that "everything exists somewhere". Or, in other words, as previously argued...the Universe being infinite is a necessary condition, but not a sufficient condition to ensure that any/every event that has a finite probability must occur/exist somewhere in the Universe.

[A. Neumaier]

This stems from the exact same arguments as in quantum field theory: there has to be some high-energy cutoff.

Can you show why it should follow from that???


Now you're mixing different descriptions of the same system. But it isn't true in any event. The computation of entropy has to count the full set of microstates, which for real particles also includes things like spin and angular momentum, as well as energy.

If you look at the books of statistical mechanics, you'll find that microstates means only ''energy eigenstate'', and not ''arbitrary state''.

[Chalnoth]

Can you show why it should follow from that???
If the integration over macrostates is limited at some high energy, and every component of that integration is finite (that is, if the function is well-defined everywhere), then it will have to be finite, because it will be a representation of a sum over a finite (but large) number of states.

If you look at the books of statistical mechanics, you'll find that microstates means only ''energy eigenstate'', and not ''arbitrary state''.
I don't think this is true at all. The basis you do your sums in is completely irrelevant. It has to be, by nature of the underlying mathematics. The only reason why the sums are done in the energy basis is because:
1. Most introductory statistical mechanics books neglect complications like spin, angular momentum, and any other potential quantum numbers that are different from energy.
2. It is much easier to do the sums in terms of energy because the total energy of the system is one of the macroscopic variables we use.

In principle you could always change to some other basis, and if it's done right you have to come up with the exact same answer, but it's going to be much more difficult to connect the other basis to the macroscopic variables.

That said, this is an off-topic argument, because it simply has no application to my original statement, which had nothing whatsoever to do with entropy. Remember, I was making two separate points when talking about the finite number of potential states. The entropy argument was one argument, and is a separate one from the purely quantum-mechanical one.

The purely quantum-mechanical argument is that as long as you cut off your states at some high energy, there are a finite (though large) number of states. You came back and stated that you can also have superpositions of those states, and since there can be an infinite number of superpositions, this finite number of quantum states leads to an infinite number of possibilities.

Not so, I said, because the superpositions are merely a representational issue: any superposition of states can be represented as an eigenstate of the right operator. You'll still always get the exact same number of states, no matter the representation you use, as long as you do the counting correctly. This second argument for the finite number of states has nothing to do with statistical mechanics.

[A. Neumaier]

If the integration over macrostates is limited at some high energy, and every component of that integration is finite (that is, if the function is well-defined everywhere), then it will have to be finite, because it will be a representation of a sum over a finite (but large) number of states.

But this can be argued only locally. The energy cutoff of QFT is something at the level of individual scattering events, while the integration over macrostates in statistical mechanics never had such a cutoff.



I don't think this is true at all. The basis you do your sums in is completely irrelevant. It has to be, by nature of the underlying mathematics. The only reason why the sums are done in the energy basis is because:

No. The only reason why the sums are done in the energy basis is because the canonical ensemble involves the Hamiltonian, and the trace defining the entropy reduces to a sum _only_ in a representation where the basis states are energy eigenstates.



The purely quantum-mechanical argument is that as long as you cut off your states at some high energy, there are a finite (though large) number of states.

And I pointed out that both your hypothesis and your conclusion are flawed.

[Chalnoth]

But this can be argued only locally. The energy cutoff of QFT is something at the level of individual scattering events, while the integration over macrostates in statistical mechanics never had such a cutoff.
Typically you don't do any integration over macrostates in statistical mechanics. The integrations are over microstates. And you don't need any cutoff there because we are generally considering systems that are at such low temperatures that any cutoff that would come in from high-energy physics is irrelevant.

But when considering all possible states of the system, you have to integrate the number of states over the ensemble of all possible macrostates. As long as the number of states for any given macrostate is finite, and as long as you have to cut off your integral at some energy (so that the integral doesn't go to infinite), the result also has to be finite.

No. The only reason why the sums are done in the energy basis is because the canonical ensemble involves the Hamiltonian, and the trace defining the entropy reduces to a sum _only_ in a representation where the basis states are energy eigenstates.
And the reason why the canonical ensemble includes the Hamiltonian is because energy is one of the macroscopic variables. It is the only operator used because in the classical treatment, energy is the only thing that is allowed to be mixed (the particle number and volume tend to be fixed). When considering more complicated systems, such as a quantum system including spin or one where the particle number is allowed to vary, you have to make the ensemble a bit more complicated, so that it incorporates these added degrees of freedom.

It doesn't really matter, though. You can still transform to another basis if you like. The results will necessarily come out the same. It's just that the math will be horribly difficult, and thus it's much easier to just remain in the eigenbasis of your ensemble.

And I pointed out that both your hypothesis and your conclusion are flawed.
No, because you changed topics and started talking about statistical mechanics in an argument that had nothing to do with statistical mechanics.

[A. Neumaier]

No, because you changed topics and started talking about statistical mechanics in an argument that had nothing to do with statistical mechanics.

As if entropy and counting quantum states could be done without statistical mechanics.

[Chalnoth]

As if entropy and counting quantum states could be done without statistical mechanics.
Huh? Counting states is a component of statistical mechanics, but hardly requires it. Entropy doesn't even need to come into the argument when all you're interested in is the total number of possible states.

[Terraqueous]

About how long did it take for the quark-gluon plasma to cool?

I'm not completely sure on this, but I think the answer is 10^-6 seconds.

[GODISMYSHADOW]

This is undecidable.

Suppose you'd record every detail about the history of the universe, wait till it has died, and then replay it in a perfect simulation (where of course, everything is already there). The physical laws would be exactly the same - without the slightest detectable difference.

Are you suggesting the universe is a simulation?

[A. Neumaier]

Are you suggesting the universe is a simulation?

No, only that we couldn't distinguish it experimentally from a simulation. The physical laws would be exactly the same if the simulation was perfect.

[Tanelorn]

"No, only that we couldn't distinguish it experimentally from a simulation. The physical laws would be exactly the same if the simulation was perfect."


I am not sure I understand what you are actually saying there but I can say that there is a great difference between a mathematical simulation on a computer with a cpu executing single arithmetic instructions one bit at a time and the space time reality we are part of. In a similar way it is highly unlikely that life like intelligence can ever be created on such a simple calculating device.

[A. Neumaier]

"No, only that we couldn't distinguish it experimentally from a simulation. The physical laws would be exactly the same if the simulation was perfect."


I am not sure I understand what you are actually saying there but I can say that there is a great difference between a mathematical simulation on a computer with a cpu executing single arithmetic instructions one bit at a time and the space time reality we are part of. In a similar way it is highly unlikely that life like intelligence can ever be created on such a simple calculating device.

Of course. If our universe were a simulation, it would have been simulated on one of God's hyper-computers with a very different physics and technology.

The point is, we couldn't see the difference in the results.

[Tanelorn]

Or the Universe and God could be one and the same thing. No simulation required :)

[GODISMYSHADOW]

Of course. If our universe were a simulation, it would have been simulated on one of God's hyper-computers with a very different physics and technology.

The point is, we couldn't see the difference in the results.

So many different views! To me, the universe is a probability with no
provable objective reality.

[Tanelorn]

Or the Universe and God could be one and the same thing. No simulation required :)



After hearing Anthony Hopkins discuss his support yesterday of the Philosopher Spinoza's views I decided to dig a little deeper and was pleasantly surprised that I share many of the sentiments:


Albert Einstein named Spinoza as the philosopher who exerted the most influence on his world view (Weltanschauung). Spinoza equated God (infinite substance) with Nature, consistent with Einstein's belief in an impersonal deity. In 1929, Einstein was asked in a telegram by Rabbi Herbert S. Goldstein whether he believed in God. Einstein responded by telegram: "I believe in Spinoza's God who reveals himself in the orderly harmony of what exists, not in a God who concerns himself with the fates and actions of human beings." Spinoza's pantheism has also influenced environmental theory; Arne Næss, the father of the deep ecology movement, acknowledged Spinoza as an important inspiration.


http://en.wikipedia.org/wiki/Baruch_Spinoza


I apologise if this is overly philosophical, I will not add to this, I just thought it was an interesting comment.

[Chalnoth]

Yeah, I personally never liked that idea as it always seemed to me that "God" carried with it far too much anthropomorphic meaning to be anything but misunderstood when used in that way. It sounds like an attempt to re-purpose the religious word to describe some feeling of awe or wonder regarding the universe itself. But I just don't see the purpose in doing that. Can't we describe the majesty of the universe without resorting to anthropomorphic words? And there remains, to me, a significant downside in that the religious merely use it as an excuse to trumpet their own views (the religious absolutely love to imagine that science is on their side, and famous scientists talking about "God" are exceptionally tantalizing).

[Tanelorn]

Chalnoth, I sympathise with your views also. In fact I find I can move between Atheism, Agnosticism and Pantheism, sometimes all on the same day. Perhaps in his statement Einstein was helping by leading people from the old superstition anthropomorphic based religions into a higher state of enlightment, taking baby steps so to speak. Hopefully we will avoid the fate that Sagan was so concerned about. The main reason I have for sometimes believing in something greater is that it sometimes appears to me that there was a very powerful and intentional force behind the creation of the universe. It cant be proven, but the universe seems so finely tuned, too much so for random chance. The whole thing seems so unlikely, and instead we could have had a universe consisting of nothing more than an infinite amount of green jelly!

In the Anthony Hopkins interview, a fellow Welshman, I particularly agreed with his views regarding people "who know the truth". Such certainties gave rise to people like Hitler with plans for everyone. I have come to similar conclusions myself.

[Deuterium2H]

For me, the biggest surprise was that an empty set has a cardinality of 1. (Did I say this right?) This just pissed me off, and got me reading about vacuous truth, and it wasn't long before I threw my hands up in exasperation and stopped trying to understand why.

But because of my frustration, I didn't like the joke "in a set of zero mathematicians, any one of them can do it [change a light bulb]." I actually remarked, to no one in particular, that "in a set of zero mathematicians, three of them are actually tomatoes." I liked this better because, "Hey, if we're being ridiculous, let's just let it all hang out and be ridiculous." What can I say, I was annoyed and was on that previously described tomato kick at the time.

But whatever, I accept on faith alone that an empty set is actually "one," because Wikipedia told me so... but I don't have to like it.

But all in all, I really, really like Set Theory, because as I said, with it, it seems possible to describe just about anything at all using math.


Congratulations, to both you and Chalnoth. I now completely agree with that statement. Gold star for youse guys. Although I'm thinking, as I said before, that I never really disagreed, I just didn't understand what infinity actually meant (I thought it literally meant "exhaustive.")

Hi Sage,
Sorry for resurrecting this older thread, but I happened to be re-reading through it for another reason, and had previously missed a statement you made, in error, that may cause all sorts of confusion if left uncorrected. The Cardinality of the Empty Set (Null Set) is not one, it is zero. The Set that contains the Empty Set is equal in Cardinality to one. In fact, in axiomatic Set Theory (e.g. ZFC), the existence of the Empty Set is defined as fundamental Axiom. It is upon this, and the following Pair Set and Sum Set axioms that larger Sets are created...thusly:

{ }= ø = 0
{{ }} = {ø} = 1
{{{ }}} = {ø,{ø}} = {0,1} = 2
{{{{ }}}} = {ø,{ø},{ø,{ø}}} = {0,1,2} = 3
etc., etc.

[nakian]

I read some answers that tended to argue that the possibility that everything could exist was unlikely. Other comments gave the impression that having a twin in another world sounded like sci-fi... Maybe you should spend some times reading what Max Tengmark has to say about the Multiverse http://space.mit.edu/home/tegmark/PDF/multiverse_sciam.pdf. Also find out more here http://en.wikipedia.org/wiki/Multiverse#Level_I:_Beyond_our_cosmological_horizo n.

The argument Tengmark makes is that worlds similar to ours are very likely, that is the likelihood that you have a twin somewhere in another world is high. Those un-observable universes , those of level-I, that is worlds beyond our cosmological horizon, will probably be of an infinite number. They will all have the same physical laws and constants as ours. Everything that is possible in our world will be possible in those worlds. In that sense, everyhing that could happen here, even if it will never happen here or has never happened here, would probably have happened or will probably happen somewhere in a Level-I un-observable world. In conclusion, it is highly probable that you have a twin somewhere, dating J-Lo's twin in that world...

[nakian]

Albert Einstein named Spinoza as the philosopher who exerted the most influence on his world view (Weltanschauung). Spinoza equated God (infinite substance) with Nature, consistent with Einstein's belief in an impersonal deity. interesting comment.

By impersonal do we mean "unconscious"? Because, Spinoza God also possesses the Attribute of being infinitely conscious. I am not to sure what being infinitely conscious means, but I pretty sure it's not the same thing as "unconscious". Am I confusing things here?

[Tanelorn]

nakian, welcome to PF!

Also I thank you for the multiverse links and question about Spinoza.

These subjects are very interesting to me, however they are also highly speculative so we may need to discuss them elsewhere. This Cosmology forum is meant for questions on the hard science of the standard model, but the thread seems to have survived thus far.

This paper on Spinoza is interesting. On pages 23 and 24 there is discussion on Spinoza's view of conciousness:
http://philosophy.fas.nyu.edu/docs/IO/2575/garrett.pdf

I presume you also enjoy the works of Nakian?

[Tanelorn]

By impersonal do we mean "unconscious"? Because, Spinoza God also possesses the Attribute of being infinitely conscious. I am not to sure what being infinitely conscious means, but I pretty sure it's not the same thing as "unconscious". Am I confusing things here?

By "impersonal deity" Einstein could be meaning one or more of the following.

1. not personal; without reference or connection to a particular person: an impersonal remark.
2. having no personality; devoid of human character or traits: an impersonal deity.
3. lacking human emotion or warmth: an impersonal manner.

I agree though that Spinoza's God is infinitely conscious, whereas Einstein seems to be saying that his God is devoid of human character, traits and personality.

More recently some may have also relegated God further, to a God of nature, an unconscious force of creation. Lovelock's Gaia principle may also be related to this view of God. ie. A Gaiaverse.