On Wednesday, 26 February 2014 22:19:35 UTC+1, Virgil wrote:
> If one does not have something like an axiom of choice, there is no > > guarantee that any infinite set need have any well-orderable subsets.
The axiom of choice is no guarantee either. It is impossible to well-order the uncountable set of real numbers. Well-ordering implies distinguishing and that means indexing. The axiom of choice only proves that matheologians are immune against contradictions. AC "proves" that there are uncountably many indexes. Set theory with and without AC proves that there are at most countably many indexes. Contraditction for everybody except the masters of dispelling.