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Normally, when people say "the universe is infinite" they generally mean something like "the observable universe is locally flat, it is easy to assume that it is flat way beyond the observable universe, I don't know what lies beyond that, I won't talk about that, that is enough for practical purposes" (or any other proposition implying infinity in any direction and assumed for simplicity's sake while actually focussing on a small region of the space, the observable universe). So, "the universe is infinite" does not seriously mean that physical space goes on forever because this simply makes no sense at all, "infinite" is not a number, it is not a magnitude, it is not a property of real things (if you disagree with this premise please explain how the notion of infinity can be seriously applied to any direction of physical space, i.e. how you can seriously think to extend, sic et simpliciter, such a property of your mathematical model to the real world). (Personally I don't think this should be counted as a theory of the shape of the universe at all).

On the contrary, "the universe is finite and unbounded, compact, closed in all directions" is indeed a meaningful and in principle acceptable theory because it can be thought, described, discussed for what it actually implies. Of course, we don't know what lies beyond the observable universe but the idea of a compact universe makes sense and it seems to me this is the only meaningful explanation currently available of what "could" be (it is not necessarily the right, true explanation, of course).

Is there any other meaningful alternative (which can be thought, described, discussed)? Is there any alternative to the dichotomy "finite universe" vs "infinite universe"? Is there anybody out there working on models that overcome this dichotomy?

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I disagree with your premise. First, the universe being infinite does not imply anything at all about its spatial curvature without additional assumptions. Second, even with the usual cosmological assumptions of large-scale homogeneity and isotropy, being infinite would not imply spatial flatness, as negative curvature would still be possible.

On the contrary, "the universe is finite and unbounded, compact, closed in all directions" is indeed a meaningful and acceptable theory.

This this alternative still makes grand assumptions about the structure of the universe beyond what we can observe, I don't see why you attribute more 'meaning' or 'acceptability' to it. At the end of the day, all we really know is that the bit of the universe around us is very close to being flat. Any statement whatsoever about the larger structure of the universe is going to introduce some assumptions not backed by observations, and you haven't provided much guidance to what kind are 'meaningful and acceptable'.

Is there any other meaningful alternative?

You seem to be using that word in an idiosyncratic sense I'm not familiar with. In any case, you're mistaken about what "the universe is infinite" implies.

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I have edited and clarified my premise: "the universe is infinite" does not seriously mean that physical space goes on forever because this simply makes no sense at all: it just means "I don't know what lies beyond" and this is not a theory, it just states the awareness of our lack of knowledge. "The universe is compact" is meaningful because it is actually offering a logic proposition which can be thought and described and discussed in the sense of the logic of propositions (i.e. a logic proposition is a proposition that makes sense whether true or false). –  randomatlabuser May 18 at 20:23
    
@randomatlabuser: I'm sorry, but I find your premise absolutely bizarre. "The universe is infinite" means the universe is not finite. It says nothing about knowledge; it simply a claim that might be right or wrong--whether or not we know a proposition is distinct from what that that proposition means. And this one is completely sensible. –  Stan Liou May 18 at 21:04
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@randomatlabuser: I increasingly doubt that your question has anything to do with astronomy, but rather some philosophical crusade against "infinity". What we actually know is a conditional: if the universe is (a) isotropic, (b) homogeneous, (c) not positively curved, (d) general relativity is correct, then it is infinite. That's a matter of logical deduction. Any or even all of those premises may be mistaken, and one can examine how justified they are in turn, but that's a completely different question then whether the conclusion is (semantically or logically) meaningful. –  Stan Liou May 18 at 21:28
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@StanLiou As far as I know the universe is flat with 0.44% error. But isn't it possible that our observable universe is only a small part of the whole universe and hence appears to be flat? –  Yashbhatt May 19 at 4:06
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Sure. It's possible that the universe is either positively or negatively curved and is only approximately flat on the scale we can see; the former case breaks (c). It's possible that the universe has a wildly varying curvature on scales greater than we can see; that would mean homogeneity is a mistake. It's even possible for it to be flat yet still be finite and compact; that would mean isotropy is out. One can question/deny any or all of those assumptions, and some scientific theories do this explicitly (e.g., eternal inflation). It just doesn't have anything to do with being "meaningful". –  Stan Liou May 19 at 20:37

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