# Is the Universe Infinite or Finite?

Is the Universe Infinite or Finite? I was having that debate with a friend of mine today. My position was that it is finite because it is expanding. My logic was that the fact that is expanding indicates that it must be finite, because how could something be infinite and getting bigger?

His position was that it was infinite though he couldn't really explain why he thought that. (to give you some context, he also believes his daily horoscope offers him meaningful guidance and that aliens built the pyramids).

You may have guessed by now that neither one of us are physicists, so to settle the argument, I proposed that we contact an expert.

Confident of my correctness, I suggested to my buddy that we should have friend of this here site @seanmcarroll (the brilliant physicist and author of "From Eternity to Here") give us an answer.

I tweeted at Sean the subject line of this thread and he was kind enough to tweet me back in a matter of minutes. He said:

"Nobody knows. Sometimes that's the answer."

He then tweeted

"There may not even be an answer. de Sitter space (e.g.) can be finite or infinite depending on how you slice it."

So not wanting to badger Sean with follow up questions, I ask you: If we know the Universe is continually expanding (which to me suggests that it has a finite size that is increasing), how can it be labeled as anything other than finite?

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### Replies to This Discussion

Something can be infinite and still be expanding. For example, take the set of positive odd integers. There are an infinite number of them, so the sie of the set is infinite. Now add the number 2 to that set. It has increased in size by one, and is still infinite. Now add all of the rest of the positive even integers. The set has increased by an infinite amount and is still infinite. You can continue to make the set larger by adding negative integers, fractions, etc, and it will keep expanding, yet at every point it is infinite in sie.

It's a common error to think of 'infinite' as a set size\value.

Holy Cantor, Batman!

Very well explained. Thank you.

There are different sizes of infinities.

The examples you give will not increase the size of an infinite set of numbers.

Look up real numbers and the difference between rational and irrational numbers.

Infinity is a mathematical concept and has no real world (universe) application. Paradoxes like the one you mentioned are evidence of that.

Strangely, infinity has a few things in common with what may at first appear to be the opposite concept: zero. Twice infinity is infinity; twice zero is zero. Half infinity is infinity; half zero is zero. And on and on.

Both concepts belong in math not in real world discussions.

Hey Dave and Mike,

This is correct. However, there is no such thing as an infinite physical quantity. Physicists reject theories or conclusions that result in infinite physical quantities because they are meaningless. For example, the GR tensor gives infinite eigen values inside black holes and GR is said to "break down" at that point for that reason.

So, an infinitely sized universe is, in fact, physically meaningless. The universe is finite. Einstein referred to it as "finite yet unbounded". What he meant was that there is a curvature of space such that if you travel far enough in one direction you will eventually return to where you started, just as if you were traveling on a 3-D sphere, but in this case it is 4-D space-time.

Therefore, you are correct (it is finite) and your friend is wrong.

- kk

Unfortunately for your argument the latest finding is that the universe is flat on a large scale, not curved.  I get this from watching lectures given by an astronomer; I Am Not An Astronomer.  The same astronomer offered no conclusive statement on the size of the whole universe other than it was at least as large as the observable universe.

Also, the bit about inside black holes approaching a singularity is, I think (and this is just me, not me relaying someone else's verbiage--if there is a real astrophysicist in the room, please feel free to chime in!) is due to the fact that it is assumed that once the material that makes the black hole collapses into the black hole, nothing whatsoever holds it up and it all collapses into a literal point of zero radius.  I am not sure why they make that assumption; I should think the only thing necessary is that it be smaller than the event horizon.  (Another possibility that occurs to me would be that it does indeed collapse to a point, but takes literally forever to do so, and it's therefore not a point mass... yet.  It's asymptotic.)

Hey StevenInCO,

Nice points. But I guess the point I'm making is that physics theory and fact is not mathematics; the two things do not commute identically. What may be meaningful in mathematics is not necessarily meaningful in nature. Infinity is one such construct that, while well defined mathematically, is utterly meaningless in nature.

The breakdown of GR in black holes is fundamental: the GR breakdown is due to the fact that GR is not a complete theory of nature and cannot properly characterize all types of natural forces (strong nuclear, weak nuclear and em). So, the *reason* you are seeking for this is simply that GR is incomplete. It is no more complicated than that.

In order to know the more fundamental reason for this breakdwon one would need to be able to characterize all the forces of nature by deduction from a single set of mathematical relations, which constitutes a unification of existing theories into one theory by whose rules all nature behaves.

- kk

Some people use the term 'universe' in reference only to our particular bubble of expanding space-time, others use it to indicate the absolute totality of anything and everything that in-fact exists. These may not be equivalent. What precisely do you mean when you say 'the universe'? Our accelerating space-time may represent only a very small fraction of what actually exists. We simply don't know. But that does not mean that we will never know or that the truth is in any way ultimately unknowable. After all, people once believed that we could never, by any means humanly possible, understand the nature of the stars. Then we developed stellar spectroscopy. Until we develop better tools, acquire better data, it seems premature to take too firm a stance on either side of the proposition. Maybe neither answer is correct and by framing the question in such a manner we are presenting ourselves with a false dichotomy?

When I say "the universe", I am referring to our ball of expanding spacetime that grew out of the big bang. It was infinitely small,dense and hot and now it is enormous, expanding and cooling. There's still a part of me that says that thing is finite.

Interestingly enough, there is a hypothesis (covered in Lauwrence Krauss' latest book A Universe From Nothing) that presents the idea that our universe is just a local difference inflationary rate and that other universes are created similarly. That there is a vast, infinite expanse of expanding spaccetime and that our universe came about due to one small segment of spacetime undergoing a period of extraordinarily rapid inflation. It's a very interesting read, I recommend it.

Hello Dave: I second your opinion of Krauss' book. Its worth buying and studying. Although its not a text book and he could have elaborated further on Cosmic Inflation. Chapter six was a bit skimpy.

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