Well-ordering and choice in toposes
This paper gives a new illustration of an interesting feature of topos theory: results or concepts equivalent in the context of sets are no more equivalent in the general context of toposes. It is a well-known result by Diaconescu that a topos verifying the axiom of choice (every epi splits) is boolean. What about Zermelo theorem on the equivalence between the axiom of choice and the fact that every set can be well-ordered? The answer is as follows: the author introduces a new definition of well-ordering, which coincides with the usual one in sets, and constructs non-boolean toposes such that every object can be well ordered. In section 2, a new definition for the axiom of choice in a topos is given, which is equivalent to the former one for sets and which does not imply booleanness as it is proved by the construction of examples.
scientific article; zbMATH DE number 4051857