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Is it conceivable that something finite can become infinite? Isn't there an inherent conceptual problem in a transition from finiteness to infinity? (My question comes from science, but the scientists don't seem to bother to explain this, such as in the case of gravity within a black hole -- a massive star collapses into a black hole and gravity in it rises to infinity? The more interesting example to me is the notion that the universe may well be infinite, but the main view in cosmology is that it began as finite and even had a definable size early on in its expansion. How could an expanding universe at some point cross over to have infinite dimensions?)

A few comments on Hilbert's Hotel (since Charles Taliaferro has brought that up) and "actual infinities": If you want a standard presentation of the usual Hilbert's Hotel "paradox", which has nothing to do with money, then check out Wikipedia's good entry . The "paradox" just dramatizes the basic fact that an infinite set can be put in one-one correspondence with a proper subset of itself. There is nothing paradoxical about that: on the contrary, it is tantamount to a definition of what it is for a set to be (Dedekind) infinite. Can there be "actual infinities" in the sense of realizations of Dedekind infinite sets in the actual world? Well, money won't do, to be sure (but that's just a fact about money, not about the general impossibility of "actual infinities"). Suppose you think that there are space-time points, and that actual space-time is dense -- i.e. between any two points there is another one. Then the points in a space-time interval will be Dedekind infinite. [Proof: label the end...

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