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

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The discussion centers on whether an infinite universe implies that all possibilities exist somewhere, with participants debating the nature of infinity and its implications. While some argue that an infinite set does not encompass all possibilities, others reference quantum mechanics to suggest that the potential for existence may lead to actualization. The conversation also touches on the lack of consensus regarding the universe's finiteness and the complexities of defining "infinite." Additionally, participants explore hypothetical scenarios involving parallel worlds and the divergence of experiences. Ultimately, the dialogue emphasizes the need for precise definitions when discussing concepts like infinity and existence.
  • #61
Chalnoth said:
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.
 
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  • #62
Dmitry67 said:
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.

Dmitry67 said:
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.
 
  • #63
Dmitry67 said:
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).

Dmitry67 said:
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
 
  • #64
Chalnoth said:
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?
 
  • #65
Dmitry67 said:
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.

Dmitry67 said:
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.
 
  • #66
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.
 
  • #67
Dmitry67 said:
At first, could you clarify what do you mean by "math isn't science"? Do you mean that we can't 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).
 
  • #68
Chalnoth said:
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 can't perceive.
 
  • #69
Entropee said:
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 can't 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.
 
  • #70
Im going to look that up that sounds really interesting.
 
  • #71
Entropee said:
Im going to 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.
 
  • #72
rasp said:
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

https://www.amazon.com/dp/0375422277/?tag=pfamazon01-20

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.
 
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  • #73
It looks like I need to return to this topic. :smile: 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
 
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  • #74
Thanks Chalnoth I actually couldn't find it on my own lol
 
  • #75
rasp said:
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.
 
  • #76
Axuality said:
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).

Axuality said:
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.
 
  • #77
Chalnoth said:
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. :)
 
  • #78
The problem, Axuality, is that you're abusing what it means to define a word.

First, as I said, there is never anyone 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".
 
  • #79
Chalnoth said:
The problem, Axuality, is that you're abusing what it means to define a word.

First, as I said, there is never anyone 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. :)
 
  • #80
JnWaco said:
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.
 
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  • #81
Deuterium2H said:
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".

Deuterium2H said:
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.
 
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  • #82
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 irrelevant 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.
 
  • #83
Surely there are exotic elements we know nothing about that would allow for seemingly improbable situations :wink:
 
  • #84
Godswitch said:
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".
 
  • #85
Deuterium2H said:
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.

Deuterium2H said:
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.

Deuterium2H said:
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 irrelevant 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.
 
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  • #86
xantox said:
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).

xantox said:
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.
 
  • #87
Chalnoth said:
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.
 
  • #88
xantox said:
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.
 
  • #89
Deuterium2H said:
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.
 
  • #90
Chalnoth said:
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.
 

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