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

[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 infinite 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 separated 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 possibility 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 [insert vague dimensions that I know nothing about here]? 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 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:

https://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 batgarbage 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, anyone 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 occurred 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 possibility 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:

https://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:

https://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.