Talk:Axiom of Choice

Not really controversial anymore

Any proof which uses the axiom of choice can be transformed into a proof that doesn't. Granted, it will be a somewhat more complicated proof, but it always works, and that's a fact. That is the reason that AC is much less controversial these days than it was, in the early 1900s.

There is a complete explanation of the process and the proof that it's reliable here.

Also, the profoundly intuitive trichotomy is equivalent to AC, so be careful what you call controversial. BenjB 20:29, 27 January 2008 (EST)

Actually, the Axiom of Choice has been proven independent of ZF, so there is no such transformation of a proof. Otherwise, "prove" AC as follows:

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1. Axiom of Choice | Reason: Axiom of Choice

Then transform it to not need AC. Result: AC proven in ZF,so ZFC=ZF. But AC proved independent of ZF. Therefore, no such transformation exists. QED SamSamson 21:50, 8 June 2008 (EDT)

This axiom actually can prove things which are (provably) unprovable without it; an example is Zorn's lemma. (Funnily enough, although Zorn is a guy's name, it's also a German word meaning "rage" ;-) --AMackenzie 14:29, 2 August 2008 (EDT)

The Axiom of Choice can also be used to "prove" absurdities, as explained by the entry here.--Aschlafly 14:52, 2 August 2008 (EDT)

It is a wonderful thing in mathematics when an absurdity is proven - it expands the consciousness. Either the "absurdity" isn't as absurd as once thought (the Earth is a ball, not a plain; there are just as many whole numbers as there are fractions), or some axiom needs to be rethought, or you've made a mistake. Whatever, you can learn from it.--AMackenzie 17:45, 2 August 2008 (EDT)

Banach-Tarski isn't absurd; it's just counterintuitive. -CSGuy 22:21, 2 August 2008 (EDT)