<p>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.</p>
<p>Disclaimer: The following is viewed as naughty, troublesome thinking by the Astrophysics establishment:</p>
<p>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?</p>
<p>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.</p>
<p>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.</p>
<p>In fact, space is literally the only thing we know of where the concept of infinity truly applies, for emptiness can never stop.</p>

CC BY-SA 4.0
2021-10-17T19:27:17.050Z
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
CC BY-SA 4.0
2021-10-17T20:23:35.050Z
@pela A finite universe would always be an entity with an outside; this is an inescapable reality. That dark void up there, space, never ends.
White Prime
CC BY-SA 4.0
2021-10-18T06:33:46.190Z
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"
Prallax
CC BY-SA 4.0
2021-10-18T08:12:56.143Z
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.
Alchimista
CC BY-SA 4.0
2021-10-18T13:04:22.420Z
As @Alchimista says, I definitely understand why you might think this, but it is simply not true that a finite universe implies an “outside”. The best visual analogy I can give you is the 2D surface of a 3D ball, but it’s just an analogy and is in danger of alluring you to think that a 3D universe implies the existence of a higher dimension, which it doesn’t. To fully appreciate this, you need the mathematics behind. You could ask for this as a separate question, although it might be better suited for physics.SE.
pela
CC BY-SA 4.0
2021-10-18T14:01:27.700Z
@pela But math isn't reality. Look, I'm simply asking you to strip away the theories, maths and assumptions from your mind. Space can only be a literally infinite void. I'm not even including matter here. I'm talking about space. There's no way it can end. And I know something can't go on forever; however, space is the one single thing that must.
White Prime
CC BY-SA 4.0
2021-10-18T15:19:53.190Z
@WhitePrime The limitations of your imagination are not limitations of reality, and unsupported beliefs are not knowledge.
Christopher James Huff
CC BY-SA 4.0
2021-10-18T16:12:12.363Z
@Christopher James Huff No unsupported beliefs, Christopher. I'd love to hear you describe how emptiness can just stop, or come to an end.
White Prime
CC BY-SA 4.0
2021-10-18T18:08:34.813Z
It doesn't need to. For example, the universe could well be a 3-sphere or a 3-torus, finite but with no boundaries. These possibilities don't go away just because you don't like math.
Christopher James Huff
CC BY-SA 4.0
2021-10-18T20:03:05.523Z
@Prallax With due respect, Prallax, I put it to you that you lack honour after using that shockingly inaccurate and uncharitable analogy.
White Prime
CC BY-SA 4.0
2021-10-19T08:58:37.850Z
@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
CC BY-SA 4.0
2021-10-20T14:41:05.363Z
[Infinity](https://www.youtube.com/watch?v=6HybPBUNzz8) is [weird](https://www.youtube.com/watch?v=eSgogjYj_uw)!
zucculent
CC BY-SA 4.0
2021-10-20T15:38:36.970Z
@zucculent But that video is attempting to explain infinity using numbers and thought experiments. Look, space is literally the only thing that we can say is infinite. What's more, space is the only entity, or subject, where it even makes sense to employ the term and concept of infinity.
White Prime
CC BY-SA 4.0
2021-10-20T15:49:07.550Z
@WhitePrime I should have been clearer. My intent in linking the videos was to show that there is more than one kind of infinity. It isn't as simple (or intuitive) as you're making it out to be. I wasn't trying to link it to space.
zucculent
CC BY-SA 4.0
2021-10-20T16:02:57.310Z
@zucculent Well, in any event, I'll end my infinity gambit with this: I think some people refuse to accept occam's razor with respect to this whole issue. The simplest, most organic, fresh conclusion to make is that emptiness can't stop.
White Prime
CC BY-SA 4.0
2021-10-20T16:11:47.573Z
Occam's Razor doesn't say the explanation with the fewest arbitrary assumptions *must* be the right one. It only says it's the *most likely* one.
Also, your explanation for why space would be infinite even in closed 3D space assumes that there is a whole other spacial dimension we have no access to, perhaps infinite ones, since what's stopping us from always adding one more?
zucculent
CC BY-SA 4.0
2021-10-20T16:26:16.190Z
Occam's Razor still requires that the simple explanation *work*. Given two explanations of equal explanatory power, one should favor the simpler one. However, by excluding mathematics and theory, you have stripped your claims of all explanatory power. All you have are repeated assertions that "it must be this way", with nothing to support this belief.
Christopher James Huff
CC BY-SA 4.0
2021-10-23T17:48:20.997Z
I upvoted, not because I agree, but I like your risk-taking posturing. But, one thing you claim as absolute, undeniable, truth, is that there can be only one Infinity. If there is only one infinity then it cannot contain another infinity. Would you accept that?
Ng Ph
CC BY-SA 4.0
2021-10-17T14:43:35.443Z
2021-10-17T14:43:35.443Z
White Prime
<p>Something infinite can expand.</p>
<p>Consider an infinite length of elastic.  There are (infinitely) beads attached to it at 1m gaps. You might label one of the beads &quot;0&quot;, then the next one is &quot;1&quot;, and &quot;2&quot; and so on.  Beads on the other side are labelled &quot;-1&quot;, &quot;-2&quot;... The elastic stretches along its whole length until all the beads are 2m apart.  And we will do this so that the bead labelled &quot;0&quot; stays put.  The bead &quot;1&quot; moves to where &quot;2&quot; was, the bead at &quot;2&quot; moves to where &quot;4&quot; was ...</p>
<p>Obviously this isn't possible in practice, but logically it is possible.  You never run out of space, because it is infinite.</p>
<p>So has the elastic rope expanded?  <em>I say yes</em>.</p>
<p>Is it longer than it was before?  <em>I say no</em>.</p>
<p>Is the a paradox or logically impossible?  <em>I say no</em>, because this is how infinities behave. Note that at no point have I said anything about &quot;infinity plus 1&quot;</p>
<p>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.</p>
<p>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.</p>

CC BY-SA 4.0
2021-10-19T07:19:06.190Z
You say that all beads are now 2m apart instead of 1m yet the elastic rope is not longer than before. This means half of the beads have miraculously disappeared. Where should they have disappeared to?
Thomas
CC BY-SA 4.0
2021-10-19T11:25:01.570Z
@Thomas - it doesn't mean the beads have disappeared. That is an assumption not based in reality.
Rory Alsop
CC BY-SA 4.0
2021-10-19T14:10:46.833Z
@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.
James K
CC BY-SA 4.0
2021-10-19T18:35:55.067Z
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.
Thomas
CC BY-SA 4.0
2021-10-19T20:32:47.907Z
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.
James K
CC BY-SA 4.0
2021-10-20T14:28:53.033Z
[Infinity](https://www.youtube.com/watch?v=6HybPBUNzz8) [is weird](https://www.youtube.com/watch?v=eSgogjYj_uw)!
zucculent
CC BY-SA 4.0
2021-10-20T20:12:01.933Z
@JamesK I did not say the beads disappear. I said they *would have to* disappear if your claims would be true. This is simple math: the number of beads on the rope is given by the length of the rope times the density of the beads. If the length of the rope stays the same but the density decreases (as you claim), the number of beads decreases.It is your task to resolve this paradox, not mine.
Thomas
CC BY-SA 4.0
2021-10-20T20:40:35.880Z
@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.
Christopher James Huff
CC BY-SA 4.0
2021-10-16T21:40:09.950Z
2021-10-16T22:01:43.607Z
2021-10-16T22:01:43.607Z
James K
<p>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 <span class="math-container">$n$</span>. If you're familiar with linear algebra, this looks like the matrix <span class="math-container">$$\begin{bmatrix} n&amp;0&amp;0 \\ 0&amp;n&amp;0 \\ 0&amp;0&amp;n \end{bmatrix}$$</span> The coordinate point <span class="math-container">$(1,1,1)$</span> will go to <span class="math-container">$(n,n,n)$</span>. The coordinate point <span class="math-container">$(-2,1.5,10)$</span> will go to <span class="math-container">$(-2n,1.5n,10n)$</span>.  When expanding the coordinate plane in this way, it won't be expanding <em>into</em> anything, which is what I imagine you meant by bringing up <span class="math-container">$\infty+1$</span>. The distance between different points is simply increasing.</p>
<p>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 <span class="math-container">$n$</span>, the point at <span class="math-container">$1$</span> goes to <span class="math-container">$n$</span>, and the point at <span class="math-container">$-2$</span> goes to <span class="math-container">$-2n$</span>.</p>
<p>One may be tempted by this description to thing that there is one point in space that stays in place, since <span class="math-container">$n \times 0=0$</span>, 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, <a href="https://en.wikipedia.org/wiki/Expansion_of_the_universe#/media/File:Space_Expansion_(no_center).png" rel="nofollow noreferrer">we'd never be able to tell</a>.</p>

CC BY-SA 4.0
2021-10-20T10:53:07.827Z
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
CC BY-SA 4.0
2021-10-20T14:19:55.930Z
@pela That's why I also mentioned the idea of just multiplying all coordinates by the constant $n$.
zucculent
CC BY-SA 4.0
2021-10-20T14:30:42.887Z
But you're right, a 1D version would be much simpler. I'll edit my answer.
zucculent
CC BY-SA 4.0
2021-10-22T12:09:48.053Z
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
CC BY-SA 4.0
2021-10-19T20:01:37.113Z
2021-10-20T14:36:08.493Z
2021-10-20T14:36:08.493Z
zucculent
<p>Having seen much interrogation on the term of infinite, I could try to give some insight on the concept, in this context.</p>
<p>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.</p>
<br>
<p>If this is indeed the case, maybe there is another way to look at the concept of infinite:</p>
<p>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.</p>

CC BY-SA 4.0
2021-10-19T07:10:33.043Z
A constantly expanding finite thing will always stay finite. It would only have an infinite size after an infinite time, which will never happen.
Thomas
CC BY-SA 4.0
2021-10-20T13:27:30.323Z
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.
Christopher James Huff
CC BY-SA 4.0
2021-10-23T21:45:47.923Z
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.
Ng Ph
CC BY-SA 4.0
2021-10-19T00:53:22.347Z
2021-10-20T18:54:08.440Z
2021-10-20T18:54:08.440Z
Aekzo
<p>Nice paradox! &quot;Only things that are <strong>finite</strong> in size can <strong>expand</strong>&quot;, true or false?</p>
<ul>
<li><p>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.</p>
</li>
<li><p>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 &quot;expansion&quot;).</p>
</li>
</ul>
<p>The Universe seems to expand because we observe a Doppler red-shift, from which we <strong>infer</strong> that stars must be moving further apart.</p>
<p>To be more scientifically correct, which should state: The <strong>observable Universe</strong>, 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.</p>
<hr />
<ul>
<li>EDIT1 (on <strong>apparent</strong> contradiction between Infinity and Expansion)</li>
</ul>
<p>After we retain that the &quot;Universe is infinite&quot; and the &quot;Universe is expanding&quot; are just working assumptions, can we say that they are contradictory (because &quot;infinity cannot expand&quot;, as argumented by the OP and many posters here)?</p>
<p>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 &quot;apple-line&quot;. 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 &quot;size&quot; in &quot;occupied space&quot; (and the same cardinality!).</p>
<p>&quot;Infinity&quot; and &quot;Expansion&quot; 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 &quot;occupying more space&quot;, or taking previously <strong>unoccupied</strong> 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.</p>
<p>But we cannot use this definition for Infinity, because, by our own convention of infinity: it can take an arbitrary large size.</p>
<p>And therefore, Infinity and Expansion (in the sense of density, or particles moving far apart each other) are not exclusive.</p>

CC BY-SA 4.0
2021-10-26T18:18:04.453Z
So what would then be the definition of 'expansion' that would apply in case of an infinite array?
Thomas
CC BY-SA 4.0
2021-10-26T19:48:44.293Z
@Thomas, what I propose is to define "expansion" by "increase of distance separations" between particles. This is exactly what astronomers postulate by observing the red-shift of distant stars radiations. The definition would apply both for finite and infinite things, such as your infinite array.
Ng Ph
CC BY-SA 4.0
2021-10-27T20:46:45.467Z
You said yourself that increasing the distances between particles by e.g. a factor 2 would be the same as taking out every second particle. Taking out every second particle is not an expansion though. How would you distinguish between the two cases (just looking at the original and the expanded configuration)?
Thomas
CC BY-SA 4.0
2021-10-27T22:02:28.117Z
@Thomas, let's take  {1,2,3,...}, 1 would not see 3 going into "red-shift" if 2 disappears, right? There is no **increasing of distance** between particles observed. What I was pointing out is that "occupied space" is an ill-defined concept for infinite things. An infinite array in expansion and an infinite array, not in expansion, but having every second particle removed have the same "occupied space". But only one would exhibit a red-shift, only one "expands".
Ng Ph
CC BY-SA 4.0
2021-10-30T20:22:26.780Z
Whether we can or should interpret the observed redshift as an expansion is  not the question here, but only whether an *infinite* universe can expand or not.. Now you suggest to define the expansion by an increase of the distance of particles in a given volume element. But for this, particles would have to leave a volume element and go somewhere else (increasing the density there).. Tell me where they are going if the density is supposed to  *decrease everywhere* including infinity. The particles would have to be shifted *beyond* infinity, which is a contradiction in  terms.
Thomas
CC BY-SA 4.0
2021-10-30T20:58:25.603Z
@Thomas, when you reason that, in an expansion, particles have to go "somewhere else", you are already trapping yourself in the world of finite things. Only in this world can you define the "somewhere else" of something. Else is beyond which you would have zero particles, beyond which, before expansion, you would have density=zero. And hence in expanding particles within a **boundary** have to go beyond such a boundary. Something which has a boundary is by definition finite.
Ng Ph
CC BY-SA 4.0
2021-11-02T19:09:08.953Z
You can not quantify an expansion without resorting to finite values. If all particles move from x to 2x then this is their expanded location. But this requires x to be finite. Expanding $\infty$ to 2$\infty$ would mean that there is something larger than infinity, which is a contradiction in terms.
Thomas
CC BY-SA 4.0
2021-11-02T20:43:41.937Z
@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.
Ng Ph
CC BY-SA 4.0
2021-11-06T15:01:24.960Z
I did not assume anything. It is a trivial mathematical fact that a global expansion factor (e.g. 2) would also apply in the limit of x being infinite $$\lim_{x \to \infty}\frac{2x}{x} = 2$$. So the size of the universe would also have to double if it is infinite.
Thomas
CC BY-SA 4.0
2021-11-06T16:29:47.350Z
@Thomas, big math flaw: this limit is that of a ratio of finite things. We are saying here, let's expand (say by 2) something that is **already** infinite, not some thing finite that grows, tends toward, but at any step remains finite. You know that infinite over infinte is indeterminate in math.
Ng Ph
CC BY-SA 4.0
2021-11-07T16:36:56.560Z
*You know that infinite over infinte is indeterminate in math* Exactly. That's why it is not possible to expand an infinite universe. You can only define the expansion factor for finite distances.
Thomas
CC BY-SA 4.0
2021-11-07T18:47:59.677Z
@Thomas, have I said that we need to define the expansion factor of the Universe? Current  science has taught that the Universe is expanding because the stars look like they are moving away. No one is establishing that it is expanding by any factor, because such factor would be meaningless. All arguments you have brought in to contradict this are based on concepts applicable only on finite things.
Ng Ph
CC BY-SA 4.0
2021-11-08T19:19:26.423Z
*have I said that we need to define the expansion factor of the Universe?* 
$$ $$
Yes, you implied it when you said this $$ $$
*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"** ).*
$$ $$
Density is defined as (number of particles / interval) For an infinite universe this this would result again in ($\infty/\infty$)
Thomas
CC BY-SA 4.0
2021-11-08T19:47:18.887Z
@Thomas, that's a description of a paradox. A paradox is when you arrive at a contradiction. Have you read somwhere that I (or somebody else) was claiming what the expansion ratio of the Universe is, 2? 3.14? 1000?
Ng Ph
CC BY-SA 4.0
2021-11-11T20:59:49.920Z
If your assumptions result in a paradox, then it follows logically that at least one of them must be wrong, in this case either the assumption that the universe is infinite or that it is expanding.
Thomas
CC BY-SA 4.0
2021-11-11T22:14:24.457Z
@Thomas, in a paradox both alternatives appear equally credible, yet they contradict themselves. Otherwise it is not a paradox. I don't mind that you keep  deriving properties of infinity from your comfort zone of finite things.
Ng Ph
CC BY-SA 4.0
2021-10-23T11:06:34.657Z
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.
Thomas
CC BY-SA 4.0
2021-10-23T17:40:40.400Z
@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.
Ng Ph
CC BY-SA 4.0
2021-10-24T08:53:32.373Z
I did not say anything different from what you said as far as finite things are concerned.
Thomas
CC BY-SA 4.0
2021-10-24T09:00:29.807Z
@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.
Ng Ph
CC BY-SA 4.0
2021-10-24T09:25:31.750Z
How can you not agree with it? It is trivial that if something expands into previously empty space, the density increases there (from previously zero to something finite). That is also the case for your water/sugar example: if you add more water the water level rises and the sugar density becomes finite in regions where it was previously zero, and it decreases in the original volume. Overall, the amount of sugar stays the same.
Thomas
CC BY-SA 4.0
2021-10-24T09:41:25.913Z
@Thomas, my bad. I agree.
Ng Ph
CC BY-SA 4.0
2021-10-21T20:29:45.860Z
2021-10-24T14:22:54.003Z
2021-10-24T14:22:54.003Z
Ng Ph
<p>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</p>
<p><a href="https://i.sstatic.net/XG8hZ.gif" rel="nofollow noreferrer"><img src="https://i.sstatic.net/XG8hZ.gif" alt="enter image description here" /></a></p>
<p>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.</p>
<p>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</p>
<p><a href="https://i.sstatic.net/mKS2k.gif" rel="nofollow noreferrer"><img src="https://i.sstatic.net/mKS2k.gif" alt="enter image description here" /></a></p>
<p>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).</p>
<p>The problem with an 'expansion' of an infinite distribution is therefore that particles would be lost, as the density would be smaller in <em>all</em> regions along the infinite line.</p>
<p>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</p>
<p><a href="https://i.sstatic.net/vIKa7.gif" rel="nofollow noreferrer"><img src="https://i.sstatic.net/vIKa7.gif" alt="enter image description here" /></a></p>
<p>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).</p>
<p>=========</p>
<p>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 <a href="https://wiki.lesswrong.com/wiki/Quick_reference_guide_to_the_infinite?_ga=2.194246824.409164306.1634984584-532805791.1634984584" rel="nofollow noreferrer">LessWrong Wiki</a> for an extended discussion and also the Wikipedia entry regarding <a href="https://en.wikipedia.org/wiki/Natural_density" rel="nofollow noreferrer">Natural Density</a>.
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.</p>
<p>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.</p>

CC BY-SA 4.0
2021-10-23T19:50:07.810Z
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.
Ng Ph
CC BY-SA 4.0
2021-10-24T09:08:23.147Z
@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](https://wiki.lesswrong.com/wiki/Quick_reference_guide_to_the_infinite?_ga=2.194246824.409164306.1634984584-532805791.1634984584) for an extended discussion and also the Wikipedia entry regarding [Natural Density](https://en.wikipedia.org/wiki/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
Thomas
CC BY-SA 4.0
2021-10-24T09:19:43.080Z
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.
Ng Ph
CC BY-SA 4.0
2021-10-24T14:49:36.730Z
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.
Ng Ph
CC BY-SA 4.0
2021-10-25T18:26:09.250Z
@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).
Thomas
CC BY-SA 4.0
2021-10-23T16:22:00.960Z
2021-10-24T09:41:32.943Z
2021-10-24T09:41:32.943Z
Thomas
<p>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?</p>

CC BY-SA 4.0
2021-10-27T20:39:58.527Z
@Pathfinder That Hotel paradox analogy is flawed. It is only a semi-infinite hotel as has a starting number. In a genuinely infinite hotel no rooms would become vacant by moving people into different rooms.
Thomas
CC BY-SA 4.0
2021-10-28T08:56:23.883Z
@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.
Prallax
CC BY-SA 4.0
2021-10-30T09:22:05.460Z
@Prallax 'Semi-infinite' means that the extension is infinite in one direction but bounded in the other. It is a common expression in mathematical physics (for instance in the theory of stellar atmospheres). In case of the Hotel 'paradox', the numbers start at 1, so they are bounded towards below at this point. If you would include room numbers 0 and all negative integers as well (only then you could use this as an analogy to an infinite universe), there would not be any vacant rooms created by moving all guests forward by 1.
Thomas
CC BY-SA 4.0
2021-10-31T08:38:04.127Z
@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.
Prallax
CC BY-SA 4.0
2021-11-02T19:29:52.940Z
@Prallax Telling the guests with negative numbers to stay in their rooms would be the same as not having any negative numbers at all. Again, it would correspond to a kind of 'semi-expanding' universe. Anyway, note that instead of telling the guests to move rooms, you could just as well re-number all rooms. But this means that, with the original scenario, rooms would start at 2, so there would not be a room 1 available to move somebody new in. All available rooms would still be occupied.
Thomas
CC BY-SA 4.0
2021-10-16T21:49:37.690Z
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.
notovny
CC BY-SA 4.0
2021-10-18T18:31:30.793Z
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.
Aaron F
CC BY-SA 4.0
2021-10-19T09:03:24.067Z
1/2 Agree with what you wrote in OP. It is hard to imagine infinite space that is constantly expanding. Same like it is hard to imagine that space has any physical boundaries. More practical is to imagine space like surface of expending balloon, where if you could travel faster than speed of universe is expanding and never change flight course, you should (after long !!! time) finish in the exact same place where you started. Many astronomers now believe that space is flat and infinite, because they are disappointed that they never could measure positive curvature of observable universe.
David Cage
CC BY-SA 4.0
2021-10-19T09:04:56.727Z
2/2 But that could be because observable universe is so extremely small part of entire universe, that it looks flat. Exactly like there are small places on Earth where we would measure zero curvature (or even negative curvature), despite that Earth is almost perfect sphere. Sure it impossible to verify if either finite or infinite option is true, but when Astronomers try to explain what is shape of entire universe to general public, they should always start with more practical description and then offer other one (infinite constantly expanding space) as an alternative.
David Cage
CC BY-SA 4.0
2021-10-19T13:44:49.707Z
Regarding the behaviour of ‚expanding‘ infinities, you might want to read the Hilberts Hotel https://en.m.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel :)
Pathfinder
CC BY-SA 4.0
2023-02-27T05:43:25.820Z
This is one of the best anti FTL arguments I have heard.
Kevin Kostlan
CC BY-SA 4.0
2023-08-11T09:19:47.073Z
I get what you're saying and don't myself believe the universe to be infinite because it's just too incredible to me that the concept of infinity can exist in reality, i.e. outside of maths. But it's a mistake to think that infinity as a number that's just "the highest number" and can't therefore can't expand.  

e.g. how many integers > 0 are there? Infinite. But there is plenty of room for "expansion" because you can add in the integers < 0, and after that you can add the non-integers between 0 and 1 of which there are also infinite... etc. etc.
komodosp
CC BY-SA 4.0
2023-08-11T09:24:12.197Z
Also: "Six septillion miles away, there might be a planet that figured out FTL" - unless FTL really is impossible, or the closest planet to discover it is still so far away that it still takes thousands or more years to get here...
komodosp
<universe><observable-universe><size>
Logically, how can the universe be infinite in size?
Max
# Logically, how can the universe be infinite in size?

(Asked by: username_0 on 2021-10-16)

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?

**Tags:** universe, observable-universe, size
> **Comments:**
> **username_3**: 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.  
> **username_4**: 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.  
> **username_5**: 1/2 Agree with what you wrote in OP. It is hard to imagine infinite space that is constantly expanding. Same like it is hard to imagine that space has any physical boundaries. More practical is to imagine space like surface of expending balloon, where if you could travel faster than speed of universe is expanding and never change flight course, you should (after long !!! time) finish in the exact same place where you started. Many astronomers now believe that space is flat and infinite, because they are disappointed that they never could measure positive curvature of observable universe.  
> **username_5**: 2/2 But that could be because observable universe is so extremely small part of entire universe, that it looks flat. Exactly like there are small places on Earth where we would measure zero curvature (or even negative curvature), despite that Earth is almost perfect sphere. Sure it impossible to verify if either finite or infinite option is true, but when Astronomers try to explain what is shape of entire universe to general public, they should always start with more practical description and then offer other one (infinite constantly expanding space) as an alternative.  
> **username_6**: Regarding the behaviour of ‚expanding‘ infinities, you might want to read the Hilberts Hotel https://en.m.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel :)  
> **username_1**: @Pathfinder That Hotel paradox analogy is flawed. It is only a semi-infinite hotel as has a starting number. In a genuinely infinite hotel no rooms would become vacant by moving people into different rooms.  
> **username_2**: @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.  
> **username_1**: @Prallax 'Semi-infinite' means that the extension is infinite in one direction but bounded in the other. It is a common expression in mathematical physics (for instance in the theory of stellar atmospheres). In case of the Hotel 'paradox', the numbers start at 1, so they are bounded towards below at this point. If you would include room numbers 0 and all negative integers as well (only then you could use this as an analogy to an infinite universe), there would not be any vacant rooms created by moving all guests forward by 1.  
> **username_2**: @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.  
> **username_1**: @Prallax Telling the guests with negative numbers to stay in their rooms would be the same as not having any negative numbers at all. Again, it would correspond to a kind of 'semi-expanding' universe. Anyway, note that instead of telling the guests to move rooms, you could just as well re-number all rooms. But this means that, with the original scenario, rooms would start at 2, so there would not be a room 1 available to move somebody new in. All available rooms would still be occupied.  
> **username_7**: This is one of the best anti FTL arguments I have heard.  
> **username_8**: I get what you're saying and don't myself believe the universe to be infinite because it's just too incredible to me that the concept of infinity can exist in reality, i.e. outside of maths. But it's a mistake to think that infinity as a number that's just "the highest number" and can't therefore can't expand.    e.g. how many integers > 0 are there? Infinite. But there is plenty of room for "expansion" because you can add in the integers < 0, and after that you can add the non-integers between 0 and 1 of which there are also infinite... etc. etc.  
> **username_8**: Also: "Six septillion miles away, there might be a planet that figured out FTL" - unless FTL really is impossible, or the closest planet to discover it is still so far away that it still takes thousands or more years to get here...  

---

## 6 answers

### Answer by username_9 on 2021-10-17. Score: -10 | Accepted answer

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.

> **Comments:**
> **username_10**: 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.  
> **username_9**: @pela A finite universe would always be an entity with an outside; this is an inescapable reality. That dark void up there, space, never ends.  
> **username_2**: 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"  
> **username_11**: 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.  
> **username_10**: As @Alchimista says, I definitely understand why you might think this, but it is simply not true that a finite universe implies an “outside”. The best visual analogy I can give you is the 2D surface of a 3D ball, but it’s just an analogy and is in danger of alluring you to think that a 3D universe implies the existence of a higher dimension, which it doesn’t. To fully appreciate this, you need the mathematics behind. You could ask for this as a separate question, although it might be better suited for physics.SE.  
> **username_9**: @pela But math isn't reality. Look, I'm simply asking you to strip away the theories, maths and assumptions from your mind. Space can only be a literally infinite void. I'm not even including matter here. I'm talking about space. There's no way it can end. And I know something can't go on forever; however, space is the one single thing that must.  
> **username_12**: @WhitePrime The limitations of your imagination are not limitations of reality, and unsupported beliefs are not knowledge.  
> **username_9**: @Christopher James Huff No unsupported beliefs, Christopher. I'd love to hear you describe how emptiness can just stop, or come to an end.  
> **username_12**: It doesn't need to. For example, the universe could well be a 3-sphere or a 3-torus, finite but with no boundaries. These possibilities don't go away just because you don't like math.  
> **username_9**: @Prallax With due respect, Prallax, I put it to you that you lack honour after using that shockingly inaccurate and uncharitable analogy.  
> **username_10**: @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.  
> **username_13**: [Infinity](https://www.youtube.com/watch?v=6HybPBUNzz8) is [weird](https://www.youtube.com/watch?v=eSgogjYj_uw)!  
> **username_9**: @zucculent But that video is attempting to explain infinity using numbers and thought experiments. Look, space is literally the only thing that we can say is infinite. What's more, space is the only entity, or subject, where it even makes sense to employ the term and concept of infinity.  
> **username_13**: @WhitePrime I should have been clearer. My intent in linking the videos was to show that there is more than one kind of infinity. It isn't as simple (or intuitive) as you're making it out to be. I wasn't trying to link it to space.  
> **username_9**: @zucculent Well, in any event, I'll end my infinity gambit with this: I think some people refuse to accept occam's razor with respect to this whole issue. The simplest, most organic, fresh conclusion to make is that emptiness can't stop.  
> **username_13**: Occam's Razor doesn't say the explanation with the fewest arbitrary assumptions *must* be the right one. It only says it's the *most likely* one. Also, your explanation for why space would be infinite even in closed 3D space assumes that there is a whole other spacial dimension we have no access to, perhaps infinite ones, since what's stopping us from always adding one more?  
> **username_12**: Occam's Razor still requires that the simple explanation *work*. Given two explanations of equal explanatory power, one should favor the simpler one. However, by excluding mathematics and theory, you have stripped your claims of all explanatory power. All you have are repeated assertions that "it must be this way", with nothing to support this belief.  
> **username_14**: I upvoted, not because I agree, but I like your risk-taking posturing. But, one thing you claim as absolute, undeniable, truth, is that there can be only one Infinity. If there is only one infinity then it cannot contain another infinity. Would you accept that?  

---

### Answer by username_15 on 2021-10-16. Score: 13

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.

> **Comments:**
> **username_1**: You say that all beads are now 2m apart instead of 1m yet the elastic rope is not longer than before. This means half of the beads have miraculously disappeared. Where should they have disappeared to?  
> **username_16**: @Thomas - it doesn't mean the beads have disappeared. That is an assumption not based in reality.  
> **username_15**: @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.  
> **username_1**: 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.  
> **username_15**: 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.  
> **username_13**: [Infinity](https://www.youtube.com/watch?v=6HybPBUNzz8) [is weird](https://www.youtube.com/watch?v=eSgogjYj_uw)!  
> **username_1**: @JamesK I did not say the beads disappear. I said they *would have to* disappear if your claims would be true. This is simple math: the number of beads on the rope is given by the length of the rope times the density of the beads. If the length of the rope stays the same but the density decreases (as you claim), the number of beads decreases.It is your task to resolve this paradox, not mine.  
> **username_12**: @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.  

---

### Answer by username_13 on 2021-10-19. Score: 6

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](https://en.wikipedia.org/wiki/Expansion_of_the_universe#/media/File:Space_Expansion_(no_center).png).

> **Comments:**
> **username_10**: 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.  
> **username_13**: @pela That's why I also mentioned the idea of just multiplying all coordinates by the constant $n$.  
> **username_13**: But you're right, a 1D version would be much simpler. I'll edit my answer.  
> **username_10**: 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 :)  

---

### Answer by username_17 on 2021-10-19. Score: 1

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.

> **Comments:**
> **username_1**: A constantly expanding finite thing will always stay finite. It would only have an infinite size after an infinite time, which will never happen.  
> **username_12**: 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.  
> **username_14**: 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.  

---

### Answer by username_14 on 2021-10-21. Score: 0

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.

> **Comments:**
> **username_1**: 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.  
> **username_14**: @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.  
> **username_1**: I did not say anything different from what you said as far as finite things are concerned.  
> **username_14**: @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.  
> **username_1**: How can you not agree with it? It is trivial that if something expands into previously empty space, the density increases there (from previously zero to something finite). That is also the case for your water/sugar example: if you add more water the water level rises and the sugar density becomes finite in regions where it was previously zero, and it decreases in the original volume. Overall, the amount of sugar stays the same.  
> **username_14**: @Thomas, my bad. I agree.  
> **username_1**: So what would then be the definition of 'expansion' that would apply in case of an infinite array?  
> **username_14**: @Thomas, what I propose is to define "expansion" by "increase of distance separations" between particles. This is exactly what astronomers postulate by observing the red-shift of distant stars radiations. The definition would apply both for finite and infinite things, such as your infinite array.  
> **username_1**: You said yourself that increasing the distances between particles by e.g. a factor 2 would be the same as taking out every second particle. Taking out every second particle is not an expansion though. How would you distinguish between the two cases (just looking at the original and the expanded configuration)?  
> **username_14**: @Thomas, let's take  {1,2,3,...}, 1 would not see 3 going into "red-shift" if 2 disappears, right? There is no **increasing of distance** between particles observed. What I was pointing out is that "occupied space" is an ill-defined concept for infinite things. An infinite array in expansion and an infinite array, not in expansion, but having every second particle removed have the same "occupied space". But only one would exhibit a red-shift, only one "expands".  
> **username_1**: Whether we can or should interpret the observed redshift as an expansion is  not the question here, but only whether an *infinite* universe can expand or not.. Now you suggest to define the expansion by an increase of the distance of particles in a given volume element. But for this, particles would have to leave a volume element and go somewhere else (increasing the density there).. Tell me where they are going if the density is supposed to  *decrease everywhere* including infinity. The particles would have to be shifted *beyond* infinity, which is a contradiction in  terms.  
> **username_14**: @Thomas, when you reason that, in an expansion, particles have to go "somewhere else", you are already trapping yourself in the world of finite things. Only in this world can you define the "somewhere else" of something. Else is beyond which you would have zero particles, beyond which, before expansion, you would have density=zero. And hence in expanding particles within a **boundary** have to go beyond such a boundary. Something which has a boundary is by definition finite.  
> **username_1**: You can not quantify an expansion without resorting to finite values. If all particles move from x to 2x then this is their expanded location. But this requires x to be finite. Expanding $\infty$ to 2$\infty$ would mean that there is something larger than infinity, which is a contradiction in terms.  
> **username_14**: @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.  
> **username_1**: I did not assume anything. It is a trivial mathematical fact that a global expansion factor (e.g. 2) would also apply in the limit of x being infinite $$\lim_{x \to \infty}\frac{2x}{x} = 2$$. So the size of the universe would also have to double if it is infinite.  
> **username_14**: @Thomas, big math flaw: this limit is that of a ratio of finite things. We are saying here, let's expand (say by 2) something that is **already** infinite, not some thing finite that grows, tends toward, but at any step remains finite. You know that infinite over infinte is indeterminate in math.  
> **username_1**: *You know that infinite over infinte is indeterminate in math* Exactly. That's why it is not possible to expand an infinite universe. You can only define the expansion factor for finite distances.  
> **username_14**: @Thomas, have I said that we need to define the expansion factor of the Universe? Current  science has taught that the Universe is expanding because the stars look like they are moving away. No one is establishing that it is expanding by any factor, because such factor would be meaningless. All arguments you have brought in to contradict this are based on concepts applicable only on finite things.  
> **username_1**: *have I said that we need to define the expansion factor of the Universe?*  $$ $$ Yes, you implied it when you said this $$ $$ *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"** ).* $$ $$ Density is defined as (number of particles / interval) For an infinite universe this this would result again in ($\infty/\infty$)  
> **username_14**: @Thomas, that's a description of a paradox. A paradox is when you arrive at a contradiction. Have you read somwhere that I (or somebody else) was claiming what the expansion ratio of the Universe is, 2? 3.14? 1000?  
> **username_1**: If your assumptions result in a paradox, then it follows logically that at least one of them must be wrong, in this case either the assumption that the universe is infinite or that it is expanding.  
> **username_14**: @Thomas, in a paradox both alternatives appear equally credible, yet they contradict themselves. Otherwise it is not a paradox. I don't mind that you keep  deriving properties of infinity from your comfort zone of finite things.  

---

### Answer by username_1 on 2021-10-23. Score: -1

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

[![enter image description here](https://i.sstatic.net/XG8hZ.gif)](https://i.sstatic.net/XG8hZ.gif)

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

[![enter image description here](https://i.sstatic.net/mKS2k.gif)](https://i.sstatic.net/mKS2k.gif)

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

[![enter image description here](https://i.sstatic.net/vIKa7.gif)](https://i.sstatic.net/vIKa7.gif)

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](https://wiki.lesswrong.com/wiki/Quick_reference_guide_to_the_infinite?_ga=2.194246824.409164306.1634984584-532805791.1634984584) for an extended discussion and also the Wikipedia entry regarding [Natural Density](https://en.wikipedia.org/wiki/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.

> **Comments:**
> **username_14**: 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.  
> **username_1**: @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](https://wiki.lesswrong.com/wiki/Quick_reference_guide_to_the_infinite?_ga=2.194246824.409164306.1634984584-532805791.1634984584) for an extended discussion and also the Wikipedia entry regarding [Natural Density](https://en.wikipedia.org/wiki/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  
> **username_14**: 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.  
> **username_14**: 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.  
> **username_1**: @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).  

---

