[ionfish]

AC is equivalent to a lot of things. There's a collection of them on the Wikipedia page.

http://en.wikipedia.org/wiki/Axiom_of_choice#Equivalents

Something I find pretty interesting is that some of these equivalences break down in weak systems.

http://www.math.uchicago.edu/~antonio/RM11/RM%20talks/mummer...

[pndmnm]

Yup, I did a few projects on equivalents of AC back in the day. That's just my favorite "appeal to intuition" one (my favorite "appeal to intuition" against AC is: the identity function is the sum of two periodic functions (though this is a consequence and not equivalent)).

Equivalence breakdown in alternate systems is a wonderful topic. I've been trying for a couple years now to figure out how to get back into set theory now that I'm out of academia. Maybe later this summer...