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?)

Good question, and a controversial topic! Some philosophers, going back to Aristotle, are happy with the concept of a potential infinite: a series that expands indefinitely. But they are unhappy with the concept of an actual infinite, partly due to the supposition that an actual infinitude could never be attained through any number of succesive events / acts. Start now, and no matter how many events transpire it seems that (just as there is no greatest possible number) you would never reach infinity. There are abundant puzzles, going back to Zeno in the fifth century BCE, about achieving an actual infinite. Here are two brief ones, the first is called Hilbert's hotel. Imagine (for the sake of argument) that you have an infinite number of rooms in a hotel and each person pays you $50 per night. How much money do you bring in per day? An infinite amount. But now imagine guests in rooms divisible by 1,000 all check out (guest in room 1,000 checks out, guest in 2,000 check out...). How much less money will you be taking in after the check out? There will be no less money, for you will still get an infinite amount of money even though you lost an infinite amount of money. Now, imagine every guest in rooms divided by 3 check out. Same result. Buissiness is still just as good as ever. For some, this counter-intuitive result lends support for questioning the reality of actual infinites. Consider the paradox of the diary. Imagine Pat has always existed and has always been keeping a diary. But, perhaps like James Joyce, it takes her/him one year to write about one day. Would Pat be done with his/her diary today? We seem to be in the awkward position of having to say 'yes' because for every day, Pat has had a year to write about it, but then it also seems (intuitively) that Pat would be hopelessly behind (if Pat started today there would be no time at which Pat would catch up to the day in which Pat is writing).

There are responses to such paradoxes (some distinguish between larger and smaller infinities) though I am inclined to be skeptical. Could it be that infinitude (like negative numbers) has a theoretically well defined role in mathematics, but not in the actual world when it comes to actual infinites (though potential infinites are not problematic)?

A few comments on Hilbert's Hotel (since Charles Taliaferro has brought that up) and "actual infinities":

  1. 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.
  2. 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 points 0, A. By the denseness hypothesis there is a point between, label it 1. By the denseness hypothesis again there is a point between 0 and 1, label it 2. By the denseness hypothesis again there is a point between 0 and 2, label it 3. Keep on going. That gives you a sequence of points 0, 1, 2, 3 ... in the interval. And by the Hilbert's Hotel shift, mapping the point labelled n to the point labelled n+ 1, it is Dedekind infinite (for that maps the labelled set of points one-one into a proper subset of itself).] But there isn't anything in the least paradoxical about holding that there are space-time points, and they are dense.
  3. Not all Dedekind infinite sets can be put into one-one correspondence with each other, by Cantor's diagonal argument. That means there not all infinite sets are the "same size" and we can talk about smaller and larger infinities. But this hasn't anything to do with the Hilbert's Hotel "paradox" (if you are muddled about that, in its original form a "paradox" about the smallest kind of infinite set, you certainly can't unmuddle yourself by talking about larger, uncountable, infinities). We might, however, raise questions about whether sufficiently large higher infinities can be realized in a physical world at all like ours
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