# Logically, how can the universe be infinite in size?

Many people have told me that the “universe doesn’t care what you think” in my regards to it being infinite in size, and I know something that seems logical doesn’t mean anything when measured by physics yet I don’t see how the universe could be infinite and yet constantly expanding. I believe it is constantly expanding but something that is infinite in size cannot be measured and you cannot just say it is now Infinite+1 and Infinite+2. Infinity is infinity. If the universe was infinite it wouldn’t need to expand anymore. It’s as big as it ever could and will be in that case. Also the universe cannot be infinite because if it was truly infinite, every possible occurrence would have happened. Six septillion miles away, there might be a planet that figured out FTL and possibly we would have seen some aliens by now. Finally, if something was infinite in size; it would have no beginning and no end. It would simply be. Yet the cosmic background radiation proves that the Big Bang probably occurred at some point within ~20 billion years ago. Yet many of the brightest minds, Einstein, William James Sidis believed it was infinite in size. Why do people think this?

• A universe can be infinite and FTL travel can still be impossible everywhere within it. Infinite in size doesn't require every conceivable possibility. The set of prime numbers is infinite in size. None of them are $\pi$, or the letter "Q", or George Jetson. Oct 16 '21 at 21:49
• I think you're asking the right question in the wrong way. More simply put, the question is "if the universe was a finite size at the moment of the big bang, then how can it have become infinite?". One problem is that, according to general relativity, at time = 0, the universe had an infinite temperature and density. Another problem is that the universe is immeasurable, so we can't know whether it's finite or infinite. Oct 18 '21 at 18:31
• Regarding the behaviour of ‚expanding‘ infinities, you might want to read the Hilberts Hotel en.m.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel :) Oct 19 '21 at 13:44
• @Thomas what do you mean by "semi-infinite"? To me it seems like a funny concept, since half of infinite is also infinite. Would you say that the natural numbers are not infinite but just semi-infinite? And that only the Integers are truly infinite instead? I can assure you that, by the accepted mathematical definition of infinite, the natural numbers are infinite. Oct 28 '21 at 8:56
• @Thomas even if the rooms where labeled with both positive and negative numbers, I see no problem in moving every guest in a positive room by +1 and tell the guests in the negative rooms to stay in their room. This would achieve having an empty room. Oct 31 '21 at 8:38

Something infinite can expand.

Consider an infinite length of elastic. There are (infinitely) beads attached to it at 1m gaps. You might label one of the beads "0", then the next one is "1", and "2" and so on. Beads on the other side are labelled "-1", "-2"... The elastic stretches along its whole length until all the beads are 2m apart. And we will do this so that the bead labelled "0" stays put. The bead "1" moves to where "2" was, the bead at "2" moves to where "4" was ...

Obviously this isn't possible in practice, but logically it is possible. You never run out of space, because it is infinite.

So has the elastic rope expanded? I say yes.

Is it longer than it was before? I say no.

Is the a paradox or logically impossible? I say no, because this is how infinities behave. Note that at no point have I said anything about "infinity plus 1"

There is nothing logically impossible with an infinite universe. It may not be true, but logically it is fine. Perhaps everything that can happen does happen... But if FTL can't happen, then we won't be visited.

Something infinite can have a beginning. Just imagine my infinite elastic rope again. But now cut it in half. It has a beginning (at the point where you cut it) but no end. A 2d sheet can have beginning and end in two directions, but be infinite in other directions.

• @Thomas - it doesn't mean the beads have disappeared. That is an assumption not based in reality. Oct 19 '21 at 11:25
• @Thomas No, all the beads are still there. No beads have disappeared. No beads have been created. You can ask me about any bead and I'll tell you were it was before and where it is after. That should convince you that no bead have disappeared. So go ahead: pick a bead. Oct 19 '21 at 14:10
• If the length of the rope has not changed (your claim) but the distance between the beads has become larger (also your claim), then beads must have disappeared. Otherwise, one of your claims must be incorrect. This is a strictly logical conclusion. Oct 19 '21 at 18:35
• Wrong. And I challenge you again Which bead has disappeared? Give me any number. Give me the number of any bead that you think has disappeared! I say you can't do this. Oct 19 '21 at 20:32
• @Thomas you are trying to treat infinities as just really big finite numbers. This doesn't work. There is no paradox, just a failure to correctly handle infinities on your part. Oct 20 '21 at 20:40

Being infinite doesn't keep something from expanding. The expansion of space can be thought of as similar to the stretching of a coordinate plane by multiplying every coordinate by a constant, which we'll call $$n$$. If you're familiar with linear algebra, this looks like the matrix $$\begin{bmatrix} n&0&0 \\ 0&n&0 \\ 0&0&n \end{bmatrix}$$ The coordinate point $$(1,1,1)$$ will go to $$(n,n,n)$$. The coordinate point $$(-2,1.5,10)$$ will go to $$(-2n,1.5n,10n)$$. When expanding the coordinate plane in this way, it won't be expanding into anything, which is what I imagine you meant by bringing up $$\infty+1$$. The distance between different points is simply increasing.

For a simpler case, let's just imagine 1D space as a number line. When we multiply every point on this line by the constant $$n$$, the point at $$1$$ goes to $$n$$, and the point at $$-2$$ goes to $$-2n$$.

One may be tempted by this description to thing that there is one point in space that stays in place, since $$n \times 0=0$$, but that would require an objective, absolute coordinate system. That's where this analogy begins to fail. Even if there was an absolute coordinate system with an absolute center, we'd never be able to tell.

• I think this answer is closest to getting to the core of the OP's skepticism. The key sentence, if I understand their skepticism correctly, is "it won't be expanding into anything". Judging from how the question is phrased, however, I think perhaps they're not familiar with linear algebra. A simpler explanation would be the 1D version with the physical distance $d$ scaling with the comoving distance ("coordinate") $\chi$ as $d=a\chi$, where $a$ is the (time-dependent) scale factor.
– pela
Oct 20 '21 at 10:53
• @pela That's why I also mentioned the idea of just multiplying all coordinates by the constant $n$. Oct 20 '21 at 14:19
• But you're right, a 1D version would be much simpler. I'll edit my answer. Oct 20 '21 at 14:30
• Yes, it's completely correct, I just meant that it would be simpler to use just the $y = ax$ version instead of matrices and all, given my assumption of the OP's level. But it's all good :)
– pela
Oct 22 '21 at 12:09

Having seen much interrogation on the term of infinite, I could try to give some insight on the concept, in this context.

It's possible that, most often and perhaps you did, 'infinite' is mainly understood as being 'omnipresent' (and maybe as 'unlimited'), as in already being everywhere - thus raising the confusion on how it could have a boundary, in this setting.

If this is indeed the case, maybe there is another way to look at the concept of infinite:

In its first and most literal sense, the term means a thing that is not finite, being composed by in- and -finite. In that way, we can imagine something, in the context of spatial dimension, that is not being finite. In case of the Universe as we know it, it's speculated (and a scientific consensus) that it is constantly in expansion, a concept for which 'infinite' (non-finite) would fits perfectly.

• A constantly expanding finite thing will always stay finite. It would only have an infinite size after an infinite time, which will never happen. Oct 19 '21 at 7:10
• It is mathematically possible for something to expand forever, but to do so at an ever-decreasing rate so that any finite portion only approaches a finite size even after an infinite amount of time. It actually can do this whether the whole is finite or not. Oct 20 '21 at 13:27
• Well, infinite in size is simply: it is larger than any size you can imagine. Said differently, when you can't put a limit on something, you must assume it is unlimited. Nobody can prove that the Universe is constantly expanding. The part of the Universe that we can observe looks as if it is expanding. Oct 23 '21 at 21:45

Nice paradox! "Only things that are finite in size can expand", true or false?

• TRUE: because if you can't compare the size now to the size then, you can't determine the expansion rate. You can't even affirm whether that thing is expanding, shrinking, or remain in the same size.

• FALSE: because if you can compare the density of particles now to the density then, you can say whether it is expanding or shrinking. You don't need the concept of finite size to define expansion. Both finite and infinite things can expand (in this definition of "expansion").

The Universe seems to expand because we observe a Doppler red-shift, from which we infer that stars must be moving further apart.

To be more scientifically correct, which should state: The observable Universe, which is finite by definition, displays a Doppler red-shift which, with our current knowledge can be explained only by its expansion. Until we can verify that the whole Universe shows the same phenomenon, we can't conclude anything about its finiteness, or otherwise.

• EDIT1 (on apparent contradiction between Infinity and Expansion)

After we retain that the "Universe is infinite" and the "Universe is expanding" are just working assumptions, can we say that they are contradictory (because "infinity cannot expand", as argumented by the OP and many posters here)?

Let's imagine a line in space, starting where I am sitting here, and put one apple every meter on that line. If we have an infinite number of apples, having done the job, we could say: that's the size of (=space occupied by) my "apple-line". Now, let's instead space these apples 2 meters apart. Can we say that we have doubled the occupied size of the space-line? Of course not! Because, alternatively I can remove the apples at odd positions on the original apple-line, therefore keeping the same "size" in "occupied space" (and the same cardinality!).

"Infinity" and "Expansion" look contradictory only when we use the bad definition of expansion for it, the one we are used to with with finite things. We define expansion for a finite thing as "occupying more space", or taking previously unoccupied space. Hence, for a finite thing we can clearly define the space occupied and unoccupied. This is exactly as saying that we know its bound, its limit, its finite size.

But we cannot use this definition for Infinity, because, by our own convention of infinity: it can take an arbitrary large size.

And therefore, Infinity and Expansion (in the sense of density, or particles moving far apart each other) are not exclusive.

• If a finite thing expands, some of its particles will move into a region of previously unoccupied space, lowering the density in the original region but increasing the density in the new region, thus leaving the total number of particles unchanged. For an infinite expanding object there is no unoccupied space it could move into. The density would decrease everywhere, which means matter would continuously be lost. Oct 23 '21 at 11:06
• @Thomas, a finite thing can expand and increase its size (and remain finite). Your first sentence is a flawed argument. For an infinite and expanding thing, "occupied space" is undefined by definition. Your 2nd argument is therefore also flawed. If i put 100g of sugar in 1 liter of water, then add another liter, sugar density has decreased by, but I still have 100g of sugar. Replace sugar with matter and water with empty space, your 3rd argument doesn't hold water. Oct 23 '21 at 17:40
• I did not say anything different from what you said as far as finite things are concerned. Oct 24 '21 at 8:53
• @Thomas, you said that, when a finite things expands, it must increase the density in "the new region, thus leaving the total number of particles unchanged". I do not agree with that. Oct 24 '21 at 9:00
• @Thomas, the flaw in your reasoning is that you assume that, if you expand something infinitely large, you would get a larger infinity. By such reasoning, you are applying the algebra you are used to from manipulating finite things. In college, you have certainly been taught that: ∞ + ∞ = ∞. Having said that, and without trying to confuse you, there are infinities that are larger than other infinities (Aleph0<Aleph1< ...). But this is another story. We are talking here about Infinity in Aleph0. Nov 2 '21 at 20:43

If we want to give a definitive answer the to OP's question we need a concise definition what is meant by 'expansion'. Otherwise we will just end up in hand-waving philosophical arguments. Generally speaking, it seems to be clear that the process of 'expansion' must imply that the object occupies a progressively larger volume of space whilst progressively decreasing its density (in order for the total number of particles to stay the same). For a one-dimensional finite scenario this may be illustrated by the following figure

The black dots show the original positions of 4 particles here, whereas the red dots show the positions of the same particles with the distance between them increased by a factor 2 (in mathematical terms this is a linear scale transformation by a factor 2). As is obvious, the decreased density of the particles is offset by the fact that the aggregate has partially expanded into an area of previously empty space.

It is obvious that this picture is not applicable in case of an initial array which is infinitely extended, as there is no unoccupied empty space to start with. Instead, we would be looking at a picture like this

I have created this animated gif in order to simulate a scan along the whole extension of the infinite distribution and it shows that the density of the scale transformed distribution (red) is everywhere smaller by a factor 1/2. There is no region of space where there are only red but no black dots to make up for the smaller density. If you count the black and red dots going through the field of view, the ratio will always be 2:1 even if you keep on counting for an infinite time (of course, you don't have to count for an infinite time to verify this, because you know this is actually just a finite image in a loop, which, assuming a homogeneous distribution, simulates an infinite line).

The problem with an 'expansion' of an infinite distribution is therefore that particles would be lost, as the density would be smaller in all regions along the infinite line.

The important point here is that an infinite distribution is a closed distribution with regard to scale transformations. The scale transformed infinite distribution has no elements outside this distribution, unlike a finite distribution. In this sense, we could as well do the scale transformation in a closed box, which would give the following result

Here the expansion is again a factor 2 involving a reflection at the left side of the box. Because no particles can leave the box, the density of the red dots is the same as that of the black dots, in contrast to an infinite closed distribution (in this case, the locations even turn out to be the same, although the particles are exchanged).

=========

EDIT: for those trying to bring in the cardinality of sets here, note that this is not always an appropriate measure of the size of sets. See this LessWrong Wiki for an extended discussion and also the Wikipedia entry regarding Natural Density. You always have to choose an appropriate mathematical procedure for your problem, otherwise you might end up with wrong answers like the ancients did with Zeno's paradox.

Now I am not a mathematician by trade and could not give you a formal mathematical proof for the present problem in terms of set theory (yet), but I think the obvious fact that the expansion of an infinite array of points leads to a decrease of density everywhere on the infinite line (as illustrated above) should speak for itself here.

• I have upvoted because I think the argument is interesting, albeit flawed. Your statement that expansion of an infinity (of particles) is impossible because density is lowered everywhere (true), and therefore particles are lost (false). Take the set of natural numbers S: 0,1,2,3,.... Take the subset of even numbers S_e:0,2,6,8 ... As it is obtained by removing the odd numbers from S, we can agree that density has been divided by 2. Yet S and S_e have the same "size" (cardinal in Math) since I can obtain any element of S by dividing a corresponding element of S_e by 2. Oct 23 '21 at 19:50
• @NgPh Cardinality is not the only way to measure the size of sets. In some cases, depending on the problem, it may not be the appropriate way. See this LessWrong Wiki for an extended discussion and also the Wikipedia entry regarding Natural Density. You always have to choose an appropriate mathematical procedure for your problem, otherwise you might end up with wrong answers like the ancients did with Zeno's paradox Oct 24 '21 at 9:08
• I can accept that. But when you argue that when density is lowered particles are lost, you didn't mention which mathematical or physical law brings you to this statement. There are many examples where there is no loss. Oct 24 '21 at 9:19
• In fact, your demonstration that Infinity cannot expand is based on the statement that, if it does, somewhere, density must increase. You said "this speaks for itself". However, to show that it is "self-evident", that we must accept that as truth, you used an example based on a finite thing. In short, you establish a rule that "speaks for itself" for finite things, then you argue (arbitrarily) that such rule speaks for itself for Infinity. Oct 24 '21 at 14:49
• @NgPh If you do a scale transformation with a scale factor >1, the overall size of the object increases but the density decreases (not increases as you said above) such as to preserve the total number of particles. For an infinitely extended object on the other hand the overall size can not increase, so the only effect is the density decreasing everywhere i.e. particles would be lost (note that in the animated graphic you could as well get the red distribution by just taking out every second of the black points). Oct 25 '21 at 18:26

I'm taking a risk here; a risk of acquiring many down votes and ruining my very modest site score. But I shall proceed in the name of free-thinking.

Disclaimer: The following is viewed as naughty, troublesome thinking by the Astrophysics establishment:

Logically, how can space NOT be infinite, @Max? Do you somehow think that the infinite black void that is space suddenly stops? How can emptiness just end?

You see many explanations regarding this matter which include assertions that, for example, the universe may be a 'form that bends into itself'. The thing is, though, every single claim of this nature STILL ends with a shape, a form that is surrounded by infinite space. There's no escaping it.

The respected Stack user @James K says here that 'something infinite can expand'. But this, like all the fantastical claims of multiple types of infinity, miss one violently evident truth. And that is this: Infinity means literally infinite. There is only one infinity.

In fact, space is literally the only thing we know of where the concept of infinity truly applies, for emptiness can never stop.

• Sorry, but this is wrong. Not because space is or isn't infinite (we don't know), but because finite and infinite universes are both possible. You might not be able to visualize a finite universe without thinking "there must be something outside", but that does not make a finite universe impossible. A finite universe is not necessarily embedded in a larger space or hyperspace, nor does it have a boundary. Similarly, the surface of Earth does have an edge, yet it is not infinite.
– pela
Oct 17 '21 at 19:27
• What you have written is not viewed as naughty or troublesome, but just wrong. Why are you posting it if you know that the people that study the subject consider it wrong? It's like I went to the Farmer SE and said "I know you guys consider it bad practice, but in the name of free thinking I must say that pouring salt on your crops really helps them grow fast and strong" Oct 18 '21 at 6:33
• I am with Pela, although I understand that even if the Universe is finite, we the humans will naturally continue to think that something finite is embedded in something bigger. That is because we are not dealing with absolutely nothing (no space even) since evolution started. It is likely true that cosmology will always pose philosophical troubles or will remain a wonder. Oct 18 '21 at 8:12
• @WhitePrime The limitations of your imagination are not limitations of reality, and unsupported beliefs are not knowledge. Oct 18 '21 at 15:19
• @WhitePrime Math isn't reality, but it's a pretty good language to describe reality. You ask me to strip away theories, math, and assumptions from my mind, but I'm not going to do that, because what's left then is no longer science, but imagination. There's nothing wrong with imagination, but it's not going to teach you anything about the physical world. The reason is, you're free to imagine anything, and hence also imagine things that are wrong. If you wish to filter your wrong ideas from your right ideas, you need to propose a falsifiable hypothesis, and your idea doesn't do that, sorry.
– pela
Oct 19 '21 at 8:58