(October 2, 2013 at 2:08 am)Categories+Sheaves Wrote: 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
- 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?
I think this misses the point. But it does get close.
The resolving clarification here would be the domain of our discussion. It's not mathematics, nor the ontology of mathematical objects, but metaphysics, ie the meeting point of mathematics and the world we live in.
We have no problem dealing with infinites in mathematics, where we can cope with the shenanigans. I'll grant that. Finitism and ultrafinitism isn't relevant here because (as I understand them), they deal with the ontology and not with the question of actual infinites in the real world..
This is provided you've caught up on what we were talking about earlier in the other thread.
The question is whether actual infinites can instantiate in the real world, which is more a matter of the juncture of mathematics and metaphysics than pure mathematics.
