Logically, how can the universe be infinite in size?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

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• 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.

Answer 5

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).

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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.

Answer 6

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.