Difference between revisions of "Talk:Axiom of Choice"
(New page: == 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 i...) |
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Also, the profoundly intuitive [http://en.wikipedia.org/wiki/Trichotomy_%28mathematics%29 trichotomy] is equivalent to AC, so be careful what you call controversial. [[User:BenjB|BenjB]] 20:29, 27 January 2008 (EST) | Also, the profoundly intuitive [http://en.wikipedia.org/wiki/Trichotomy_%28mathematics%29 trichotomy] is equivalent to AC, so be careful what you call controversial. [[User:BenjB|BenjB]] 20:29, 27 January 2008 (EST) | ||
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| + | 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: | ||
| + | ----------------------------------------------- | ||
| + | 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 | ||
| + | [[User:SamSamson|SamSamson]] 21:50, 8 June 2008 (EDT) | ||
Revision as of 01:50, June 9, 2008
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:
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)