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How to prove the following conclusion:

[For any infinite set $S$,there exists a bijection $f:S\to S \times S$] implies the Axiom of choice.

Can you give a proof without the theory of ordinal numbers.

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    As pointed out on MathOverflow, what does it mean "without the theory of ordinal numbers"? Also, why do you wish to avoid it? If it is due to lack of knowledge on the topic, the proof does not use much of the theory of ordinals, nor I think that a proof directly trying to find a choice function would be any simpler.2011-08-09

2 Answers 2