Replies to This Discussion

Permalink Reply by Graham E. Lau on February 9, 2010 at 3:29pm

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?

Permalink Reply by Michel-san on February 9, 2010 at 7:11pm

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!

Permalink Reply by Graham E. Lau on February 10, 2010 at 2:07pm

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...