Mathematics with the negation of AC
Clearly Very important results in Math require the Axiom of choice, for example "any vector space has a base". But in the absence of AC (i.e., only in ZF) it is possible that a vector space has no basis.
In another direction append the negation of AC to ZF. What happens to algebra or analysis now ? If you know any Theorem in this new system (other than those that can be drived in ZF alone) please let me know. A fine reference would also be helpful.