The Paradox of Hilbert's Hotel

[Vincenzo Vinny G.]

We've been talking about infinites lately, but nobody wants to bite.

So let's have some fun.

http://www.youtube.com/watch?v=faQBrAQ87l4

Food for thought: Can infinites exist in the real world? This is not a trick question. Some philosophers are willing to embrace the absurdities and accept actual infinites as a part of reality. While others philosophers don't. What do you think?

[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

• 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?

[Vincenzo Vinny G.]

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.

[Categories+Sheaves]

I can dig it. Although there's a fair amount of funkiness when it comes to the notion of infinity. From the Levinas on my bookshelf:

...To be sure, things, mathematical and moral notions are also, according to Descartes, presented to us through their ideas, and are distinct from them. But the idea of infinity is exceptional in that its ideatum surpasses its idea, whereas for the things the total coincidence of their "objective" and "formal" realities is not precluded... ...The distance that separates ideatum and idea here constitutes the content of the ideatum itself. Infinity is characteristic of a transcendent being as transcendent; the infinite is the absolutely other. The transcendent is the sole ideatum of which there can be only an idea in us; it is infinitely removed from its idea, that is, exterior, because it is infinite...

I'm sure you see the problem here: if the 'metaphysical' notion of infinity is defined by its 'surpassing' instantiated things, 'actual infinity' is an immediate contradiction. Which, of course, is the reason why it has this status as a properly meta-physical concept . Non-meta-physicists (i.e. physicists) work with infinity all the time (even some of the weirder infinitary stuff, cf. Feynman Integrals) without touching the 'meta' baggage (because if they did, they wouldn't be physicists).

When I bring math into these sorts of discussions, I normally tell a joking story that starts with the angle Cantor took, and how he managed to work with infinities in a way that made them circumscribed, tame, etc. The punchline is more or less that the metaphysicians' notion of infinity seems to strike back when we look at whether the ordinals to form a set or have a cardinality associated with them. And so we have concessions like the notions of a proper class, etc. This, I assume, is at least somewhat related to your bringing up ordinals earlier.

Bleh! So much recap! Anyway, my point boils down to something like this:

• There are both 'meta' and 'non-meta' practices in the way people discuss infinity
  
• If we're discussing 'actual infinity' with the metaphysical baggage included, we need to disrupt the delimitations surrounding both types of practices
  
• We may or may not have our work cut out for us
  

So: You need to get really disruptive! It's the only way to stay on topic!

[Vincenzo Vinny G.]

I see the baggage as making it that much simpler to conclude that actual infinities are problematic to actualize. Even the kind of infinites physicists deal with I find are typically theoretical in nature.

Something I'm curious about though- of all the mathematical applications of infinities, which do you think are the closest to reality?

[Cato]

Something I'm curious about though- of all the mathematical applications of infinities, which do you think are the closest to reality?

Vinny = Infinitely stupid