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On the surface, this may seem like physics or cosmology, but I suspect my question has more to do with a misapplication of math. If it seems like a bad fit here, yell at me and I'll move it.

Let us suppose that outer space continues on without end, containing an infinite amount of mass and energy arranged in every possible permutation. Suppose also that the laws of physics are universal and invariant, and that there are no (or a negligible amount of) inaccessible regions of space due to ongoing inflation or similar.

Lastly, suppose that it turns out that some form of FTL travel is possible, be it wormholes or exotic matter powering an Alcubierre drive or whatever. Or to be more precise, suppose that via any mechanism, it is or will be possible to travel arbitrarily far in any direction you like in such a way that you are able to escape what we consider to be the observable universe. No entities are ultimately bound by their original light cones.

Granted it seems unlikely that we live in such a reality, but as far as I know, it's not absolutely out of the question. It would ensure that infinitely much life will be out there, so let's say Alice reasons there's got to be an alien very similar to her whom she'd like to meet, and if the two of them are alike enough, the alien would feel the same way about her. Independently, a similar-minded alien named Bob reaches the same conclusion.

Both Alice and Bob decide to find each other, and to be thorough about it, they upload to Von Neumann probes (classic Bob) and spread themselves out in every direction from their respective systems of origin. They travel at an effective rate greater than that at which the universe is expanding, if applicable.

So given all the painfully hand-wavy preamble, we reach my point of confusion; on the one hand, it seems inevitable that Alice and Bob will eventually make contact (i.e. their spheres of expansion will intersect). After all, they both started at essentially random points in space, and it seems like their origins must be a finite distance from each other, however far that may be, and so they should meet up after some finite amount of time.

Having said that, it also feels like nonsense. In particular, it seems like I may just be asking a convoluted version of something approximately like "Suppose you select two random integers, equally distributed over all possible integers; is the difference between them finite?" I know enough to know that selecting "random" integers in that way is some combination of ill-defined and impossible.

My question: Will Alice and Bob ever meet each other? More to the point, I'm wondering whether the two cases I've presented differ in any significant way. While this is ill-posed in the simpler numerical case I gave, it seems harder to justify if it's grounded in physics.

Does this mean that one of the assumptions I stipulated must be untrue on purely mathematical grounds? Or perhaps the universe does run on real numbers instead of discrete units, and the higher cardinality somehow rescues things. Or maybe this would be physical evidence that the Axiom of Choice is true in reality, in that the universe can "pick" two random matching people out of an infinity of options. Any insight or considered reaction would be helpful, and in lieu of receiving a concrete answer, I will accept the most useful reply.

Asaf Karagila
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Trevor
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    The distance between any two points in Euclidean space is finite. – Karl Oct 07 '22 at 03:50
  • the axiom of choice is not needed to pick finitely many members from a non-empty set – Atticus Stonestrom Oct 07 '22 at 03:51
  • @AtticusStonestrom My understanding was you needed it to pick finitely many members from an infinite set of indistinguishable elements, or something very roughly like that... – Trevor Oct 07 '22 at 03:54
  • @Karl So if one accepts my stipulations, the answer is yes, they will definitely meet? – Trevor Oct 07 '22 at 03:56
  • "Eventually" seems to be doing a lot of work here. – JonathanZ Oct 07 '22 at 03:56
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    @Trevor no, there's no need for axiom of choice to do that – that's just "existential instantiation". see for example Asaf Karagila's answer here or Carl Mummert's answer here for some discussion. the axiom of choice is only ever needed in situations where you need to make infinitely many choices – Atticus Stonestrom Oct 07 '22 at 04:00
  • @AtticusStonestrom Thanks for the link, will read up some more... I should point out though, given my assumptions, I would think anything that occurs will occur at infinitely many times and locations with pairs of Alice-and-Bobs, something I maybe shouldn't have omitted from the question... – Trevor Oct 07 '22 at 04:04

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"Randomness" doesn't enter into it one way or another; as long as you're willing to stipulate that space still looks like ordinary Euclidean space $\mathbb{R}^3$ then it is just true that if Alice and Bob inhabit two points in space then they are a finite distance away from each other. There is no need to discuss a probability distribution over such points at all.

Qiaochu Yuan
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  • Why not? In my scenario, two arbitrary entities have locations that are seemingly selected with equal distribution over all possible locations in $\mathbb{R}^3$. This seems to me equivalent to saying you're selecting two real numbers arbitrarily selected from all of $\mathbb{R}$ with a uniform distribution. So, sure, it makes sense to say that any two finite numbers will have a finite difference, but the bigger problem is that you can't make selections out of a uniform distribution on infinity (AFAIK), yet that's what the universe would seem to implicitly be doing in my scenario. – Trevor Oct 09 '22 at 19:51
  • @Trevor: you haven't specified how the universe is doing anything at all. Here's one possible model: let's suppose that the universe is divided into infinitely many equally-sized chunks, and in each chunk something random happens, drawn from some probability distribution over possible chunks of the universe. As long as an observer like Alice exists with positive probability in some such chunk, Alice exists; to be concrete let's pick a specific point in space, say the Earth, and define Alice to be the closest Alice-like observer to Earth. Similarly we can define Bob to be the closest... – Qiaochu Yuan Oct 09 '22 at 20:00
  • ...observer to Alice. These distributions are not uniform because we've defined them starting from a specific point in space. Alice exists with probability $1$ and so does Bob, and they're at a finite distance away from each other, which again is a distribution over all possible distances which is not uniform. The simplest toy model of this scenario is a collection of infinitely many coin flips, one located at each integer (as a toy model of a "random $1$-dimensional universe.") We can take Alice to be the heads closest to $0$ and Bob to be the heads closest to Alice. Is this clear? – Qiaochu Yuan Oct 09 '22 at 20:01
  • Said another way, the operation that looks like attempting to sample from a uniform distribution over an infinite universe is not Alice's generation but your attempt to select Alice out of all Alice-like observers in the infinite universe. Once you specify a well-defined way to select Alice there are no issues. – Qiaochu Yuan Oct 10 '22 at 06:19