Wednesday, February 7, 2018

A really weird place in conceptual space regarding infinity

Here’s a super-weird philosophy of infinity idea. Maybe:

  1. The countable Axiom of Choice is false,

  2. There are sets that are infinite but not Dedekind infinite, and

  3. You cannot have an actual Dedekind infinity of things, but

  4. You can have an actual non-Dedekind infinity of things.

If this were true, you could have actual infinites, but you couldn’t have Hilbert’s Hotel.

Background: A set is Dedekind-infinite if and only if it is the same cardinality as a proper subset of itself. Given the countable Axiom of Choice, one can prove that every infinite set is Dedekind infinite. But we need some version of the Axiom of Choice for the proof (assuming ZF set theory is consistent). So without the Axiom of Choice, there might be infinite but not Dedekind-infinite sets (call them “non-Dedekind infinite”). Hilbert’s Hotel depends on the fact that its rooms form a Dedekind infinity. But a non-Dedekind infinity would necessarily escape the paradox.

Granted, this is crazy. But for the sake of technical precision, it’s worth noting that the move from the impossibility of Hilbert’s Hotel to the impossibility of an actual infinite depends on further assumptions, such as the countable Axiom of Choice or some assumption about how if actual Dedekind infinities are impossible, non-Dedekind ones are as well. These further assumption are pretty plausible, so this is just a very minor point.

I think the same technical issue affects the arguments in my Infinity, Causation and Paradox book (coming out in August, I just heard). In the book I pretty freely use the countable Axiom of Choice anyway.

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