Your finite hotel has nowhere to expand to. Hilbert's hotel can accommodate any number of extra guests, including an infinite number, at any time.

Hilbert's hotel is also fully booked, recall, and a paradoxical thought experiment. Moreover, if a finite hotel has nowhere to expand to, then an infinite hotel is simply impossible.

It's not a paradox, it is a property of infinity.

If the universe is finite, then what's outside it? What does it expand into?

Finite spacetimes can expand just as well as infinite ones. There is no need for them to expand into anything and no need for anything to be outside it. Things just get farther apart, is all.
Put differently, the length of the shortest straight-line trip around the universe (analogous to circumnavigating the Earth along the equator or another Great Circle) increases.

Certainly expansion of a finite spacetime can create room for more guests if by room we just mean 'empty space'. Where the hotel & guest analogy breaks down then is with the question 'where are the extra guests going to come from?' If they were in the spacetime all along then there's no need to make room for them. On the other hand if they somehow come from outside the spacetime then the spacetime is not a closed system and we'd need a whole lot more info about what sort of system it is to begin to analyse the question.

Finite spacetimes can expand just as well as infinite ones. There is no need for them to expand into anything and no need for anything to be outside it. Things just get farther apart, is all.
Put differently, the length of the shortest straight-line trip around the universe (analogous to circumnavigating the Earth along the equator or another Great Circle) increases. — andrewk

What is the shape of a finite, flat, homogeneous, isotropic universe with dark energy?

Well, the observable universe is a sphere.

What is the shape of a finite, flat, homogeneous, isotropic universe with dark energy? — tom

According to current cosmological theories there cannot be such a spacetime. If it is flat it must be infinite.

What about a sphere?

When one says sphere, one can be referring to either

• the 2D object that is the surface or boundary of the sphere. In 3D Cartesian coordinates, this is the set of points that satisfy the equation x^2+y^2+z^2=r^2, or
• the 3D object comprised of both the boundary and the interior. In 3D Cartesian coordinates, this is the set of points that satisfy the equation x^2+y^2+z^2<=r^2

Both of these are objects that are embedded in 3D space - hence the three coordinates x, y and z.

Since space is 3D, to get something like a sphere for our universe we would need to be referring to the analog of these that are objects embedded in 4D space, which we would call 'hyperspheres' because they have one extra dimension. These would be the spaces whose coordinates satisfy the equations

• x^2+y^2+z^2+w^2=r^2 for the boundary; or
• x^2+y^2+z^2+w^2<=r^2 for the boundary and interior

The second is not possible because it has a boundary, and can be dismissed by Aristotle's ancient argument about going to the boundary and then poking my spear through it.

However the first has no boundary because it IS a boundary. It is a 3D space inhabited by 3D creatures, so they cannot point their spears in a direction perpendicular to their space. So that space is possible, and is one of the three models that is possible for the shape of the universe under current cosmological theories. Those models are elliptic, flat and hyperbolic, and the hypersphere is elliptic.

Which of the three models is the case depends on whether the overall global average spacetime curvature is positive, negative or zero. Zero corresponds to flat. We could never prove that curvature is zero because there will always be an accuracy limit to our measurements, so we can only ever determine that the curvature is less than some number. On the other hand, it IS conceivable that we could prove that curvature is positive or negative, in which case we would know the universe is elliptic (hyperspherical) or hyperbolic.

So in answer to your specific question, a (hyper)sphere is possible, but it is not flat. A spacetime that is both flat and finite would have to have a boundary, and that possibility can be dismissed by a variety of arguments, including Aristotle's, and the observation that such a space is not homogeneous (because some points are nearer the boundary than others).

Technical point. Although we always visualise a 2D sphere boundary as embedded in 3D space, it is not necessary for that to be the case. One can construct such an object without using three dimensions. Similarly, there is no need for a 3D hypersphere (which mathematicians call a '3-sphere' or just S^3) to be embedded in 4D space. I mention this otherwise pedantic-seeming point just to head off comments that then there must be something for the universe to 'expand into'.

That's a great post. Thanks.

But I do have a little point. We have these mathematical models, almost like "video games" or simulates that spit out the right predictions (they agree with observations). So we trust them, as we should, given the success so far of that approach.

But it's almost as if we have a little prediction box in some corner of the intersection of our "total" human realities and we point to that box as the real universe. I'm not saying that it's not wise to act-as-if for certain purposes, but from one perspective it looks like a little prediction machine. This is just a "map is not the territory" sort of comment, I suppose.

I really wonder why anyone would care to prove if either the universe is finite or infinite. It is beyond our capabilities to prove such a statement either way. I firmly do not believe in the big bang theory, yet, it is clearly questionable to want to prove the universe is infinite. We simply can't know for sure unless we observe its infinity. Which would take an infinity. Claims about what exists outside of this universe (which is all there ever was, is and will be) I think are also meaningless, just like determining the age of this universe, or claiming any absolute truth concerning the universe expanding or contracting or being stationary. The real question I think is why do people ask these sort of questions? What is the purpose of saying either this or that when all that is possible is mere speculation about human observations? Why do we insist on trying to "prove" matters that are completely beyond our capabilities to prove or refute (only perhaps theoretically)?

Quite, we are not aware of what this maths is, what it is showing us about what exists, existence, or how it comes to exist(etc.etc...)

"If the universe is finite, then what's outside it? What does it expand into?"

You might be in a state of delusion, or an artificial construct in a more subtle reality etc.

"If the universe is finite, then what's outside it? What does it expand into?"

You might be in a state of delusion, or an artificial construct in a more subtle reality etc. — Punshhh

If by "subtle" you mean "assuming a cognitive ability in your reader", then perhaps.

I'll try to be less subtle:

The density parameter of the universe has been measured. If the parameter = 1, the universe is flat. The result obtained is: Ω = 1.000 +/- 0.004 . The universe is flat.

We also know (via similar measurements) that the universe is homogeneous and isotropic, and expanding at an accelerating rate.

The notion that the universe is not simply connected doesn't seem to attract much attention. I suspect that if it were not, evidence would be found in CMB, and it's a strange idea anyway. The universe is not a Klein bottle or a torus.

Given that we know the universe possesses this geometry, we may now address the question of finiteness.

According to current cosmological theories there cannot be such a [finite] spacetime. If it is flat it must be infinite — andrewk

Now while andrewk's statement is true, it is not logically correct. There is no logic that prevents you from ad-hoc modifying GR to create a finite universe with a boundary, which is precisely what you are forced to do if you value finiteness more than reason. And, if you choose that course, you will be asked "what such a universe is expanding into" by someone.

I'm confused by this. Let's take the observable universe as an example. Surely it's flat (parallel lines never cross, the corners of cubes will always make right angles, etc.), finite, homogenous, isotropic, and spherical (given three spatial dimensions of equal distances)?

Yes I know what physicists have worked out. But that's besides the point, I am not questioning the scientific view of the universe. I'm pointing out that philosophically we cannot determine if, or not, our(humanity's) perception of the universe might be the result of a delusional state* and as such the perception of the finite might be a delusional mirage, or an artificial construct in a more complex and subtle reality inhabited by God like beings, or something else inconceivable to us.

This being the case we can't with certainty know what finite entails, or that there is anything finite external to our perception. It's worse than this regarding infinity, because it is entirely a figment of the human imagination.

* I am working from the observation that the only thing we can know with any certainty is that we have being and experience and that all else is secondary to this fact.

I'm confused by this. Let's take the observable universe as an example. Surely it's flat (parallel lines never cross, the corners of cubes will always make right angles, etc.), finite, homogenous, isotropic, and spherical (given three spatial dimensions of equal distances)? — Michael

We don't know any of those things. I'll consider them one by one.

Flat? The density parameter has been measured to be very close to 1, but it has to be EXACTLY 1 for the universe to be flat. But if it is exactly one there will always be values greater than and less than 1 within the error bounds of any unbiased measurement. We can never know the exact value of any measurement. If the universe is in fact flat we will never be able to tell whether that is the case or whether it is elliptic or hyperbolic with a parameter very close to 1. In the flat and hyperbolic cases the universe would be infinite. In the elliptic case it would be finite but unimaginably large.

Homogeneous and isotropic? We know these do not hold at the usual scale. The cosmological principle states that these must hold 'at the large scale' but no formal definition is given of what that means. One of my current projects is to develop a formalisation of that definition, but it keeps getting interrupted by other things. Most cosmology tends to just assume the universe is isotropic and homogeneous.

Spherical? It depends what is meant by that. Certainly we can mark out a sphere in space, and develop a spherical system of coordinates for it - indeed Spherical Coordinates is a standard type of representation of 3D Euclidean space. But that tells us nothing about the shape of the universe as a whole. The three shapes envisaged by current theories are Elliptic, Flat and Hyperbolic, corresponding to density parameters greater than, equal to and less than 1. It is conceivable we could prove the shape is elliptic or hyperbolic, because it's possible to get a confident reading about an inequality. But we could never prove it is flat because that would require a reliable measurement of an EXACT equality. The best we could say is that it is very nearly flat, but that doesn't help at all with the crucial question of whether it is finite.

Under current theories and methods, the only way to conclude the universe is finite (infinite) would be to get a measurement that shows the parameter is greater than (less than) 1, beyond the limits of experimental error.

The wiki article on the shape of the universe covers these issues and I think is rather good.

Quite, we are not aware of what this maths is, what it is showing us about what exists, existence, or how it comes to exist(etc.etc...) — Punshhh

I hear you. Lots of math has some kind of intuitive reality for me, but it's a big step from that more or less inter-subjective intuition to metaphysical statements. We also understand the "manifest image" and measurements of length and time. So science can earn our trust as a prediction machine. To me the scientific universe is a mathematical image. We place ourselves in that image, but that image is also within the much richer totality of our lives. From this perspective, any sort of reduction of all of experience to some facet of experience is nakedly absurd. It completely reason to talk about whether the universe of the scientific image is infinite or not. I'm just saying that there doesn't have to be any metaphysical investment in the question. What I'm getting at could apply to any sort of reduction, not just to the scientific image. Every metaphysical or physical theory of the totality is still just an idea in the still-untamed totality that exceeds and contains it. Every map is part of and smaller than the territory. Yet the utility of statements may depend exactly on talking and acting as if the totality has been tamed and mapped. So I'm just poking a breathing hole in the paper sky, perhaps.