# Can the Universe Be Infinite?

Some think that the universe is infinite. To convince one in the creditability of the idea they point mainly at our inability to conceive of its spatial limits. Hence, here, I use 'infinity' (of the universe) just the same way--namely, no finite number can measure it.

Imagine they are correct. Infinity of anything means that it will have both infinite spatial parts being at finite distance and infinite ones removed infinitely from one another. If so, and given the limited speed of the interaction between the objects (parts) of the universe, there would be infinite parts of the universe infinitely isolated from one another; they would 'never' interact with one another. In that case, the universe might cease to maintain its unity and thus, to hold universal laws, physics claims to study.

So, can the universe be infinite or in other words, can the idea of our inability to imagine the universe's spatial limits be viewed as a counterbalance to that of the unity (universal laws) of the universe? • Infinite universe does not imply "there would be infinite parts infinitely removed from one another". The Euclidean 3D space is infinite, but any of its parts are at finite distance from each other. Even if it had such causally isolated parts there is no problem with them sharing universal laws. Indeed, in special relativity parts of the universe are causally isolated if they are spacelike separated, but still obey the same laws. By the way, non-existence of spatial limits does not imply infinite size, spheres in any dimension have no limits, but have finite size. – Conifold May 16 '19 at 6:30
• I suggest that you clarify in your post what "infinite universe" is supposed to mean, currently it is obscure. And the universality of laws is contingent even when the regions are causally connected: there is no logical problem with the inverse square law turning into inverse cube law, for example, beyond some spatial boundary. We simply take universality as a methodological assumption for the sake of simplicity of description. If we discover that some law is not universal it is simply no longer counted as "fundamental". The drift of "fundamental" constants is actively discussed, for example. – Conifold May 16 '19 at 7:00
• @PeterJ "the idea of infinite extension is paradoxical and not plausible" Based on what, precisely? – Noah Schweber May 17 '19 at 0:45
• @PeterJ Common sense is a poor guide beyond common circumstances for which it was developed, and from which most of fundamental science and metaphysics are far removed. Bib Bang is consistent with infinite extension (of spacetime), and under current expansion projections the spacetime is infinite (expansion never stops). – Conifold May 18 '19 at 5:17
• In my opinion, the improper and careless use of the meaning of infinity underlies the issues of metaphysics, math, and physics; for example, such expression as ‘an infinite number of finite numbers’ easily makes us think that infinity is a number-like thing, when it is not. – Giorgi May 22 '19 at 12:30

## 1 Answer

There are several major assumptions here which you haven't justified (and which I don't see how to justify):

there would be infinite parts infinitely removed from one another.

The set of natural numbers is infinite, but any two natural numbers are only finitely far apart. We can have "local finitude" but "global infinitude."

given the limited speed of the interaction between the objects (parts) of the universe, there would be infinite parts of the universe infinitely isolated from one another, so that the universe is very likely to cease to maintain its unity and thus, to hold universal laws

Wy should parts of the universe infinitely far apart (even if such exist) behave significantly differently from each other? (Also, what does "unity" mean here?) And this is ignoring the possibility of physical interactions which don't depend on distance - there's nothing a priori impossibile about two objects sharing a relationship despite being far apart from each other in space.

can the idea of our impossibility of imagining the universe's spatial limits be viewed as a counterbalance to that of the unity (universal laws) of the universe

Here you seem to be conflating knowledge with truth - the universe may have fundamental properties which we can never know, and while that may be annoying there's nothing wrong with that a priori.

The question of whether it's possible for the universe to be infinite is very hard to even phrase precisely, let alone answer. But certainly I see no serious objection to that possibility here, nor am I aware of such. So I believe the answer to the title question is:

## As far as we know, yes.

• Infinity of anything means that it will have both infinite spatial parts being at finite distance and infinite ones removed infinitely from one another. To communicate at a limited speed, say c, the latter will need infinite time. They would exist like 'parallel spaces.' The problem with this, arising to physics, is that the universality of laws of the universe would become the matter of chance rather than necessity. In this sense, its subject becomes blurred. Notice too that the question opposes to each other the two ideas rather than conflates them. – Giorgi May 16 '19 at 6:54
• By the 'unity' I mean that the energy conservation law is applicable to the whole of the universe, for example. – Giorgi May 16 '19 at 12:02
• @Giorgi "and infinite ones removed infinitely from one another." ...does it now? Consider the set of all finite integers greater than 0. That set has an infinite number of members, but by definition every number in the set is finite (because that's the set we're considering... the one containing only finite numbers). It follows that the distance between any two numbers in this set is also finite. So, which is it... are you claiming this set is finite (if so, how big is it?) Or are you claiming that there are elements in this set an infinite distance from each other (which ones?) – H Walters May 17 '19 at 3:02
• @Giorgi I disagree with the notion that the set having only finite numbers in it means its size is finite. If M is its finite size, then M+1 is also a finite number, so it should contain M+1 since it has all of them. But since it contains all numbers from 1..M+1, then by definition of counting it is at least size M+1 (since we can count M+1 elements). Another way to think about it is, for any finite number M, no matter how large, this set is larger than that finite number M. So it is infinite. – H Walters May 17 '19 at 14:52
• Now consider any two numbers; hypothetically, X and Y where X<Y and X and Y are an infinite distance away. Add one more number to consider: 1; so we have 1<=X<Y, since that's a minimum. The distance from 1 to Y is Y-1 (or pred Y if you prefer). Surely the distance then from X to Y must be less than or equal to that, meaning it must be less than Y. But if Y is a finite number, that means the distance from X to Y must be finite, right? So the only way you can have X and Y be an infinite distance from each other is for Y to be infinite, and we simply don't have infinites in the set. – H Walters May 17 '19 at 14:55