Edit: I have now rewritten the question in terms of well-ordering. Before, had I used the axiom of choice directly, which made the question unclear.

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 the form that for each set a well-ordering exist. In greater detail:

• If the axiom of choice is not true, then there are usually some sets that still can be well-ordered, as for example the countable sets. All sets that can be mapped injectively into a well-ordered set can also be well-ordered. Or in terms of cardinality: All sets smaller than a well-ordered set are also well-orderable.
• So in a set theory without the axiom of choice, we have two kinds of sets: Small ones, which have a well-ordering, 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.

So the question is: What can I get instead if I cannot well-order some sets?

Edit: I have now rewritten the question in terms of well-ordering. Before, had I used the axiom of choice directly, which made the question unclear.

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 the form that for each set a well-ordering exist. In greater detail:

• If the axiom of choice is not true, then there are usually some sets that still can be well-ordered, for example the countable sets. All sets that can be mapped injectively into a well-ordered set can also be well-ordered. Or in terms of cardinality: All sets smaller than a well-ordered set are also well-orderable.
• So in a set theory without the axiom of choice, we have two kinds of sets: Small ones, which have a well-ordering, 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.

So the question is: What can I get instead if I cannot well-order some sets?

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.

Edit: I have now rewritten the question in terms of well-ordering. Before, had I used the axiom of choice directly, which made the question unclear.

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 the form that for each set a well-ordering exist. In greater detail:

• If the axiom of choice is not true, then there are usually some sets that still can be well-ordered, as for example the countable sets. All sets that can be mapped injectively into a well-ordered set can also be well-ordered. Or in terms of cardinality: All sets smaller than a well-ordered set are also well-orderable.
• So in a set theory without the axiom of choice, we have two kinds of sets: Small ones, which have a well-ordering, 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.

So the question is: What can I get instead if I cannot well-order some sets?