[DerekL]

> Until recently I never questioned the idea that, say, the positive integers and the odd positive integers are equivalent because they can be paired, but this cloning thing seems like something that falls out of that.

They really aren’t connected. The first statement (the positive integers can be partitioned into two sets, each of which has the same size as the original set) follows from the usual axioms of set theory (ZF), while the Banach–Tarski paradox cannot be proven to work without the Axiom of Choice or a similar axiom.

[nwallin]

Isn't that just because Hilbert's Hotel is a property of the natural numbers (well ordered) while Banach-Tarski is a property of the reals? (not well ordered without AoC) We can split the natural numbers into odd and even groups by starting with 1 and iterating on the odds, and starting with 2 and iterating on the evens. But because the reals are not well ordered, the step in Banach-Tarski where we pick an arbitrary point that hasn't already been grouped into a set is impossible.

The natural numbers (and therefore Hilbert's Hotel) provide a natural way to say "whatever, just pick one" but we need to invoke the well-ordering theorem (which is equivalent to the Axiom of Choice) make the same "whatever, just pick one" statement about the reals. (and therefore Banach-Tarski)