How do we know space is not infinite?

 Quote by khemist A simple way to think of infinity. Consider all the real numbers -> 1, 2, 3, 4, etc... One would certainly consider that an infinite set (give me any number, and I can get another number by simply adding 1). Now, take all the real even numbers -> 2, 4, 6, etc... That is certainly an infinite set as well, but it has half the values as the original set!
Another, and perhaps easier way to consider infinity is simply the following:

1. Infinity = the state or quality of being infinite.
2. Infinite = not finite, boundless, unlimited, indefinite, immeasurably/exceedlingly great.

e.g.: the dimensions of the universe/multiverse.

Recognitions:
Gold Member
 Quote by zeffur7 1. Infinity = the state or quality of being infinite.
a meaningless tautology

 2. Infinite = not finite, boundless, unlimited, indefinite, immeasurably/exceedlingly great.
uses words where you should use math as zeffer7 did

 e.g.: the dimensions of the universe/multiverse.
purely speculative on your part --- unproven and at present unprovable

 People following this thread might also be interested in another thread entitled "If the Universe is infinite, does that mean everything must exist somewhere". Here is a link: http://www.physicsforums.com/showpos...&postcount=101 Using Set Theory, I argue that this is not necessarily the case. Some interesting ideas discussed in that thread. I might add that this issue is stumbled over by a great number of leading physicists. Most recently, Brian Greene in his current PBS/NOVA show "The Fabric of the Cosmos". He makes the same logical error (IMHO) that is common in the infinite, "multi-verse" hypothesis...claiming that an exact duplicate of himself exists either in our infinite Universe, or in any of an infinite multiverses. The previously cited thread/link explains the pitfalls in this reasoning.

 Quote by phinds I completely understand this concept, but would be astounded if it ever turned out to be the case. I DO believe it possible that the universe has the topology that you suggest but even if it does, the thought that it could be small enough that the light would actually reach us from both directions just strikes me as so unlikely as to be silly.
This possibility has been studied by astronomers with negative results.

 Quote by Deuterium2H People following this thread might also be interested in another thread entitled "If the Universe is infinite, does that mean everything must exist somewhere". Here is a link: http://www.physicsforums.com/showpos...&postcount=101 Using Set Theory, I argue that this is not necessarily the case. Some interesting ideas discussed in that thread. I might add that this issue is stumbled over by a great number of leading physicists. Most recently, Brian Greene in his current PBS/NOVA show "The Fabric of the Cosmos". He makes the same logical error (IMHO) that is common in the infinite, "multi-verse" hypothesis...claiming that an exact duplicate of himself exists either in our infinite Universe, or in any of an infinite multiverses. The previously cited thread/link explains the pitfalls in this reasoning.
Yes, Greene is stumbling into a common fallacy.

1) I exist, therefore the probability of my existing is greater than zero.
2) In an infinite set, any event with probability greater than zero occurs infinitely many times.

The fallacy is in line 1. The probability that Greene exists could be zero.

To see this, consider the natural and real numbers. The probability that a real number is equal to a natural number is zero. Nevertheless the natural numbers exist.

So with probability zero, nevertheless the natural numbers and Greene exist.

 Patrick is it useful though to say that the probability of anything at all existing is zero? Seems like a flaw in statistics to me :)

Recognitions:
Gold Member
 Quote by PatrickPowers The probability that Greene exists could be zero.
BUT ... if it WERE zero, he would not exist and he DOES exist, so in this universe, it is NOT zero. In fact, in this universe, the probability of his existing is one.

Your argument about the numbers seems nonsensical but it could be that I'm just not following what you mean.

The satement "with probability zero ... the natural numbers ... exist" is what seems nonsensical. In a system where the probability of something existing is zero, it will NOT exist and if it does exist, then the probablity of it existing is one. I doubt you would argue with that, so what exactly ARE you arguing?

 Quote by PatrickPowers Yes, Greene is stumbling into a common fallacy. 1) I exist, therefore the probability of my existing is greater than zero. 2) In an infinite set, any event with probability greater than zero occurs infinitely many times. The fallacy is in line 1. The probability that Greene exists could be zero. To see this, consider the natural and real numbers. The probability that a real number is equal to a natural number is zero. Nevertheless the natural numbers exist. So with probability zero, nevertheless the natural numbers and Greene exist.
Hi Patrick,

No sure if you had an opportunity to read through the thread/link I provided. Not sure I agree with your beginning argument, as in this universe, we are pretty sure Brian Greene does exist, and therefore the probability of his existence is in fact one (1). However, your subsequent argument is spot on. Specifically, that there does NOT exist a one-to-one correspondence (bijection) between the Set of Natural numbers and the Set of Reals. The argument in the topic I linked to ("If the Universe is infinite, does that mean everything exists somewhere"), in summary, is that an infinite set does not imply the exhaustion of all possible "members". In other words, it is NOT a sufficient condition that an Infinite universe means that everything exists somewhere.

As a specific example, the Set of all Natural Numbers (N) has the exact same size (Cardinality) as the set of all Rational Numbers (Q). They are both infinite sets containing the exact same number of members. However, you will never find the rational number one third (1/3 = .3333) in the set N. In other words, even though N is infinite, and exactly equal in size to Q, it does not exhaust all the numbers...that is to say, there are an infinite number of Rational numbers that are not members of the infinite set N

Essentially, the same argument applies to the "infinite universe / multiverse". Just because something is infinite, does not mean it is exhaustive.

-------------------------------------------

Getting back to the second part of your post...as a further example in support of your argument:

As a thought experiment, consider a hypothetical lottery machine that contains bouncing ping pong balls, each representing a Real number...and includes all the Real numbers. When one goes to select a random ping pong ball / number (again think of the vacuum-based lottery machines), the chances of getting a Natural number is zero. In fact, the chances of getting a Rational number is also zero. Perhaps even more mind blowing, the chances of getting an Algebraic Irrational number (e.g. Square Root of 2) is also zero. Someone might ask, well what is left? When you grab that ping pong ball, you have a 100% probability of picking a Transcendental Irrational.

 Quote by phinds a meaningless tautology
Wrong. It point to the definition of infinite. That in itself is useful.

 Quote by phinds uses words where you should use math as zeffer7 did
By 'zeffer7' I imagine you errored.
The words that I used to define 'infinite' are accepted terms for defining 'infinite"--they are quite easily found in any high quality dictionary.

 Quote by phinds purely speculative on your part --- unproven and at present unprovable
"Infinity" is unproven and unprovable... what's your point??

Recognitions:
Gold Member
 Quote by zeffur7 "Infinity" is unproven and unprovable... what's your point??
I think he means that saying infinity is the dimensions of the universe/multiverse is unproven and unprovable.

 Quote by phinds BUT ... if it WERE zero, he would not exist and he DOES exist, so in this universe, it is NOT zero. In fact, in this universe, the probability of his existing is one. Your argument about the numbers seems nonsensical but it could be that I'm just not following what you mean. The satement "with probability zero ... the natural numbers ... exist" is what seems nonsensical. In a system where the probability of something existing is zero, it will NOT exist and if it does exist, then the probablity of it existing is one. I doubt you would argue with that, so what exactly ARE you arguing?
It has to do with measure theory, which is the basis of modern axiomatic probability theory. Measure theory was created to deal with infinite sets. You start out with your measure space, which is the set of all possible events, and has measure one. Then assign nonnegative measures to the subsets. The measure of a subset is equal to the probability that an event chosen from the measure space will be a member of the subset.

If the measure space is infinite then often the probability of every event is zero. Think of choosing one of the natural numbers with each number equally likely (something you can't actually do in real life.) The probability you will chose the natural number n is zero for every natural number. So if you could choose one natural number n, then even though the probability is zero nevertheless you have that number. You have proved that it was not impossible for you to chose that number. So "impossible" and "probability zero" are NOT the same thing.

Probability zero means that the probability is less than any positive number. Only zero remains. But it IS possible. (It would be silly to clutter the measure space with impossible events. What would be the point?)

Let's say you have the real numbers on the interval [0,1] and you imagine you can choose one of those real numbers with all numbers equally likely. The probability you will choose 0 is zero. But it's not impossible.

On the other hand, of you somehow DO choose zero then the conditional probability that you chose zero is 1. The conditional probability of X given X is always one. The conditional probability the Brian Greene exists given that Brian Greene exists is one.

If Brian Green were in a finite Universe then his assertion that the probability that he exists is greater than zero seems reasonable. You could theoretically count the number of planets n and then say that the chance that Greene exists is 1/n. That's more than zero. But in an infinite Universe I don't accept it. The conditional probability that Brian Greene exists given that Brian Greene exists is one, so this gives us no information other than that it is not impossible for him to exist. But as you have seen, this tells us nothing about his probability. In an infinite universe it is perfectly OK for him to have probability zero.

So now that you supposedly have gotten used to this definition, I can also say that in an infinite Universe I expect the probability that the Earth exists is zero, the probability that the visible Universe exists is zero, etc. One would expect that the bigger and more complicated something is, the lower the probability.

Now let's confuse things further. If you have an infinite set with all events equally likely, then ANY finite subset has probability zero. So even if a jillion Brian Greenes exist in an infinite Universe, his probability is STILL zero. Compared with infinity, it's insignificant. If you got in your incredible space ship and toured a million worlds a second, the probability that you would find an alien Brian Greenes would be zero. A jillion divided by infinity is still zero.

Now to really put the zap on your mind, there are plenty of infinite sets with probability zero. The prime numbers are a good example. If you pick a natural number with each such number equally likely then the chance it is prime is zero. Suppose a Brian Greene is on every prime numbered world. So if you got in your incredible space ship and toured a trillion million worlds a second for a billion years, the probability that you would find one of that infinite set of alien Brian Greenes would STILL be zero. That's how big infinity is.

Think about it for a while. Get used to it. You will realize that this has to be true, otherwise the measure of your measure space becomes infinite instead of one. And that simply will not do. Probabilities are ALWAYS 0 through 1. That's the norm.

 Quote by Drakkith I think he means that saying infinity is the dimensions of the universe/multiverse is unproven and unprovable.
Unproven certainly. But I hesitate to say unprovable. Bell came up with that theorem, so anything seems possible.

On the other hand you can say that physics never proves anything, and there is no arguing with that.

Recognitions:
Gold Member
 Quote by Drakkith I think he means that saying infinity is the dimensions of the universe/multiverse is unproven and unprovable.
yes, that's exactly what I was saying

 Recognitions: Gold Member Patrick, I can't argue with your math, but to me it seems useless (because mutually contradictory) to say that something has zero probability of occuring yet it can occur. To me it seems that the language breaks down under such nonsense regardless of what the math says.
 I've been watching Stephen Hawkings "Universe" recently and have a perspective I'd like to share. I'm not a trained scientist, just a curiosity junkie. As you mentioned, science is theorizing the universe is expanding and will eventually grow cold and dark as galaxies just continue to move apart. I'm wondering if the universe may work in a different way. Like... what if there are hot and cold areas, like the scientists say there are, and the hot areas expand while the cold areas contract, and somewhere in all this hot cold dynamic it's really staying pretty much the same size, just moving around within it's own parameters. Also, what if the hot areas (because they're expanding) kind of fold, twist and turn into their cold neighbors space, warming them up, and therefore, making the uv self renewing. what if? jus sayin. :=D
 I would like to know where, in the universe, are we. the milky way I mean. are we in the middle, in the suburbs, out on the farm? where exactly? any ideas?
 Blueyes such a situation might fall foul of Isotopy and homogeneity requirements. It is not a preferred solution to believe that somewhere, even very far beyond the observable universe, is much different to our own local space - well at least for certain scales.