"The Axiom of Choice is necessary to select a set from an infinite number of socks, but not an infinite number of shoes."


"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" 


What are people's opinions on choice?


Proofs involving choice are often delightfully unenlightening or even paradoxical, but it does have uses.

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I'm not very well versed in set theory. I've never heard of this Axiom of Choice. From what I've just been reading it sounds like something implicit in most mathematics that only requires definition in certain areas of set theory. It also seems straightforward, though once I stopped and thought about it I kind of got wrapped up in considering how the AC would be applicable to anything.

Can anyone explain the Axiom of Choice in a meaningful way?
The easiest way of seeing it outside of set theory is in linear algebra. The AC is logically equivalent to saying "every vector space has a basis".

AC can be used to construct non-measurable sets, which leads to the Banach-Tarski paradox, something that is well worth looking up!
Wow, that is intriguing. It seems to go against immediate logic; constructing infinitely many multidimensional surfaces from a single finite multidimensional surface.

There's so much about set theory that I just have no comprehension of...

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Serious discussions of mathematics

Started by Charles Daney. Last reply by Joe Morgan Dec 9, 2012. 3 Replies

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