Craig Feinstein wrote: > > Let's say I have a drawer of an infinite number of [...] > > How does modern mathematics resolve this paradox?
A few years ago Zdislav V. Kovarik made a post listing a dozen or more meaning of the word "infinity" as used in different branches of mathematics. I'm hoping that he won't mind me reposting it:
There is a long list of "infinities (with no claim to exhaustiveness): infinity of the one-point compactification of N, infinity of the one-point compactification of R, infinity of the two-point compactification of R, infinity of the one-point compactification of C, infinities of the projective extension of the plane, infinity of Lebesgue-type integration theory, infinities of the non-standard extension of R, infinities of the theory of ordinal numbers, infinities of the theory of cardinal numbers, infinity adjoined to normed spaces, whose neighborhoods are complements of relatively compact sets, infinity adjoined to normed spaces, whose neighborhoods are complements of bounded sets, infinity around absolute G-delta non-compact metric spaces, infinity in the theory of convex optimization, etc.;
each of these has a clear definition and a set of well-defined rules for handling it.
And the winner is... the really, really real infinity imagined by inexperienced debaters of foundations of mathematics; this one has the advantage that it need not be defined ("it's just there, don't you see?") and the user can switch from one set of rules to another, without warning, and without worrying about consistency, for the purpose of scoring points in idle and uneducated (at least on one side) debates.
-- When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him. Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting