The standard or “orthodox” Objectivist answer is that the universe is temporally infinite but spatially finite. Leonard Peikoff, in a podcast dated 12-29-08, said that “you can’t go outside the universe because the universe is finite and there is no out there.” He is answering a question about what happens when you reach the boundary of the universe, which he says is identical to a question he asked Ayn Rand during the first year that he knew her. The question is: If you keep going forever, wouldn’t you eventually reach the boundary, and then what happens? Peikoff clearly implies that Ayn Rand agreed with the premise that the universe does have boundaries (although she apparently gasped when he asked the question).
Here are some of the conceptual problems I have with the question Is the universe infinite?
First, what do we mean by "the universe"? Well, when I use this term, I mean "everything that exists." So the original question becomes Is everything that exists infinite?
I can't honestly say that I even know what the question is supposed to mean. The most obvious interpretation translates into the question Is there an infinite number of existents in the universe? The problem here is that "infinite," as used in this context, is not a number. On the contrary, it signifies that no specific number can be assigned. And if this is the case, then it makes no sense to speak of "everything" that exists, because there is no "every" or "all" of which we can speak.
Second, when most philosophers have spoken of an "infinite universe," they have meant that nonexistence (i.e., empty space) has no limitations. The term "space," when used in this sense , is not a thing with specific properties. It is nothing -- and nothing has no characteristics and therefore no limitations.
Confusion inevitably arises when we reify nonexistence and speak of it as if it were a type of existence. This most often occurs when we speak of space -- again, in the sense of empty space --as being "infinite." This can give the thoroughly misleading impression that "infinity" is an attribute of an existent known as "space" -- whereas all we really mean is that nonexistence has no boundaries. We can no more limit nonexistence than we can lasso nonexistence or cook nonexistence for dinner.
You argue that infinity “signifies that no specific number can be assigned. And if this is the case, then it makes no sense to speak of 'everything' that exists, because there is no 'every' or 'all' of which we can speak.”
This strikes me as confusing mathematics with physical reality. If everything means the “totality” of everything, then there should be a total we can theoretically point to. But suppose that mathematical concept (total) is not applicable to the universe as a whole, or everything which exists. Because it applies to all the things we’re familiar with, does not mean it necessarily applies to existence, as such. Again, it's the fallacy of composition.
I’m not sure what you mean when you say “nonexistence has no boundaries” or “limiting nonexistence.” I may have misunderstood you, but this seems to amount to “speaking of nonexistence as if it were a type of existence.” I have a hunch this may just be my misunderstanding of your views. In any case, let me state clearly my own view that you cannot do anything with nothing. Nonexistence does not exist. Period. And there is no such thing as empty space, because that would amount to nothing having the quality of existence—a contradiction in terms.
It is because nothing cannot exist that the idea of the universe being finite seems absurd to me. And “finite but unbounded” is equally absurd (and, based on the Sagan excerpts, apparently defies coherent explanation). If the universe is finite, it has dimensions. If it has dimensions, it has to have boundaries. And if it has boundaries, then something has to exist on both sides of those boundaries—but nothing does not exist.
Repeating the last paragraph of my response to Ted:
And once we posit [or logically imply] a boundary, we have to ask--what is beyond the boundary? Well, obviously nothing is beyond the boundary. But nothing does not exist. That's why I contend that it makes more sense to leave open the possibility that--just as time is eternal--the universe is, in fact, infinite, even though we have difficulty conceiving of what that means.