Which is equivalent to the axiom of choice. If one chooses not to accept that axiom, then the well-ordering theorem is not a theorem. One can even (consistentl

Which is equivalent to the axiom of choice. If one chooses not to accept that axiom, then the well-ordering theorem is not a theorem. One can even (consistently) append the negation of the axiom of choice to ZF, in which case this theorem is false. Mere negation of the axiom of choice is not a very interesting addition, but there are some interesting alternative axioms, such as the axiom of determinacy, which have negation of the axiom of choice as a consequence People who are fixated on definable/computable reals generally have constructivist sympathies, and many (but not all) constructivist mathematicians reject the axiom of choice. So relying on it in arguments in this context is questionable. https://mathoverflow.net/questions/119561/mathematics-with-the-negation-of-ac https://mathoverflow.net/questions/119561/mathematics-with-t...