Wikipedia's article on axiom of choice states that Tarski proved that the statement if X is infinite then X is bijectively equivalent to XxX is equivalent to axiom of choice. I know how to prove that AC implies X=XxX for infinite X. I am curious to know the proof in the other direction. Does anyone know a reference for this? It seems hard to google.
In that article wiki has a list of propositions that are weaker than the axiom of choice. Among them is the free subgroup theorem. Question: is it known that free subgroups theorem is STRICTLY weaker than AC? If so that would presumably mean that somebody has constructed a model of ZF in which free subgroup theorem is true, but AC does not generally hold.
Can anyone think of a good expository article on issues such as this related to AC?
> Wikipedia's article on axiom of choice states that Tarski proved > that the statement if X is infinite then X is bijectively equivalent > to XxX is equivalent to axiom of choice. I know how to prove that AC > implies X=XxX for infinite X. I am curious to know the proof in the > other direction. Does anyone know a reference for this? It seems > hard to google.
> In that article wiki has a list of propositions that are weaker > than the axiom of choice. Among them is the free subgroup theorem. > Question: is it known that free subgroups theorem is STRICTLY weaker > than AC? If so that would presumably mean that somebody has > constructed a model of ZF in which free subgroup theorem is true, but > AC does not generally hold.
> Can anyone think of a good expository article on issues such as > this related to AC?
> Regards, > Mike
Article? No.
Book? Try Rubin and Rubin's Equivalents of The Axiom of Choice, or a similarly titled tome.
Gerhard "Ask Me About System Design" Paseman, 2010.03.09
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