Tuesday, June 27, 2017

Set size and paradox

Some people want to be able to compare the sizes of sets in a way that respects the principle:

  1. If A is a proper subset of B, then A ≤ B but not B ≤ A.

They do this in order to escape what they think are paradoxical consequences of the Cantorian way of comparing sizes. But from one paradox they fall into another. For the following can be proved without the Axiom of Choice:

  1. If there is a transitive and reflexive relation ≤ between sets of reals (or just countable sets of reals) that satisfies (1), then the Banach-Tarski Paradox holds.
And the Banach-Tarski Paradox is arguably more paradoxical than the paradoxes of infinity that (1) is supposed to avoid.

No comments: