Response from Daniel J. Velleman on December 4, 2005

Yes, people have studied statements that contradict the Axiom of Choice. One of the most widely studied is the Axiom of Determinacy (AD). There is a Wikipedia entry that will tell you more about it.

The question of whether or not AD is consistent with the Zermelo-Frankel (ZF) axioms of set theory is a bit tricky. Of course, by Godel's Incompleteness Theorem, we can't even prove that ZF is consistent (assuming it is), so we certainly can't prove that ZF + AD is consistent. It is not even possible to prove that the consistency of ZF implies the consistency of ZF + AD. If you're willing to make a stronger assumption (the consistency of ZF + the existence of certain kinds of large infinite cardinal numbers), then you can prove the consistency of ZF + AD.

Response from Richard Heck on December 4, 2005

The Axiom of Choice (usually denoted "AC") is a statement of set theory rather than of basic mathematical logic, so the theories of interest are versions of set theory that reject AC. As Dan said, any theory containing the Axiom of Determinacy will imply not-AC, but one can also simply look at what is possible without AC and, similarly, what cannot be proven without AC. There is a nice guide to such results: Thomas Jech's The Axiom of Choice. There are also weaker forms of AC, such as the Axiom of Countable Choice (every countable set of non-empty sets has a choice function) and the Axiom of Dependent Choice (more complicated). There are also forms that are stronger than what is usually assumed in set theory, in particular, what is sometimes called the Axiom of Global Choice.