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Mathematics
Set Theory, Logic, Probability, Statistics
About the “Axiom of Dependent Choice”
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[QUOTE="andrewkirk, post: 5995190, member: 265790"] I have not heard of it, but can see why it would be useful. Sometimes in real analysis one wants to make a sequence in which the next element is different from the current one. Using the relation ##\neq## and the above axiom, one can assert the existence of such a sequence in any set with two or more elements, as the relation is entire if there are more than two elements. There was a proof that was being discussed on here the other day that needed something like that. Unfortunately I can't remember the context, other than it was real analysis - probably something about sequences. Not knowing about this axiom, I just said we had to assume AC - assuming it needed the full version. It sounds from the wiki article like this axiom is strictly weaker than AC. It would be nice if it allowed one to recover most of the popular results of real analysis without having to accept the Banach-Tarski conundrum, or the theorem that every set can be well-ordered as a conclusion. I wonder if it is weak enough to prevent either or both of those. [/QUOTE]
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Mathematics
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About the “Axiom of Dependent Choice”
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