RE: The Paradox of Hilbert's Hotel
October 2, 2013 at 2:08 am
(This post was last modified: October 2, 2013 at 2:10 am by Categories+Sheaves.)
I mean, I'm thoroughly comfortable with all the shenanigans infinity brings about... objects with embeddings into proper subsets of themselves, addition not being commutative on the ordinals, Russell's Paradox, Cantor's Paradox, all the odd things the axiom of choice lets you do, the unprovability of the continuum hypothesis in ZFC, the Lowenheim-Skolem theorem, etc. etc. etc.
The potential vs. actual infinity discussion is a pretty well-traveled road, and I don't think there's a ton of fecundity in the general terms (In what sense do we have an 'actual' infinity? Can I bet infinite money on a race? Is my sandwich infinitely divisible? Does space go on forever?). So in the spirit of keeping it on the concrete side... do you accept the principle of induction (on the natural numbers)? I ask this because
Also: when do we get to the transfinite ordinals?
The potential vs. actual infinity discussion is a pretty well-traveled road, and I don't think there's a ton of fecundity in the general terms (In what sense do we have an 'actual' infinity? Can I bet infinite money on a race? Is my sandwich infinitely divisible? Does space go on forever?). So in the spirit of keeping it on the concrete side... do you accept the principle of induction (on the natural numbers)? I ask this because
- The usefulness of induction is obvious, and mathematicians who want to throw it away (finitists, ultrafinitists) are (in a sense) shooting themselves in the foot.
- If you accept it, you're OK with dropping universal quantifiers on infinite sets (to some extent). And so the flood gates are opened!
Also: when do we get to the transfinite ordinals?
