The anti-foundation axiom is a nice thing: It not only denies the axiom of foundation, but also gives an interesting structure to the non-founded sets.

I now want to know whether there is a similar axiom that constructively denies the axiom of choice. In greater detail:

• If the axiom of choice is not true, then there are usually some sets that still have a choice function, as for example the countable sets. All sets that can be mapped injectively into such a "choice set" are also choice sets. Or in terms of cardinality: All sets smaller than a choice set are also choice sets.
• So in a set theory without the axiom of choice, we have two kinds of sets: Small ones, which have a choice function, and big ones, which do not have one. A good anti-choice axiom would therefore provide a "natural" division between small and big sets and give the big sets some additional properties that they cannot have under the usual (ZFC) set theory.