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Does it bother you that by dividing all the integers into odd and even numbers, I can end up with two infinite sets, each of the same size as the original set?

While the Banach-Tarski violates your intuitions, it is probably because it is more about how there are "multiple copies" of the real numbers in the real numbers, than anything resembling a real cut.

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Not to nitpick or assume you we're saying this, but doesn't the cardinality of the evens and odds equalling the cardinality of the integers hold even without the axiom of choice?

Curse you, iPhone keyboard and Autocorrect. s/we're/were/

Yes, the argument is more complicated, and does require the axiom of choice.

However, to me the "basic principle" is a similar argument. The important thing (for me) was to understand this is an argument about slicing up infinities in an "Infinite comb" type way, rather than anything resembling a "cut".

You might be right. The more I learn about the real numbers, the less real they seem :/

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