Freyd's models for the independence of the axiom of choice / Andreas Blass and Andre Scedrov

In 1980 P. Freyd constructed three topos-theoretic models for the independence of the axiom of choice (AC). S. MacLane asked whether these models are related to the more familiar Fraenkel-Mostowski models or the symmetric Boolean-valued models. This problem is investigated in the present paper. It is shown that the answer is positive. Of fundamental importance is the adequate definition of the notion of representation of a model of set theory by a topos. It is proved that Freyd's first example represents a Boolean-valued model obtained by first adjoining to the universe V a countable family A of mutually Cohen generic reals and then forming the submodel which is generated by V∪B∪{B}, where B is the Boolean algebra generated by A. This model is a proper submodel of Cohen's model for ZF+¬AC which is generated by V∪A∪{A}. It is shown that Freyd's topos fits into a pullback diagram of topoi where the other vertices are topoi representing related models. Analogous results are proved for Freyd's second example and the “primordial example”. The paper is carefully written and contains a description of both the category-theoretic and the set-theoretic background. (Zentralblatt)

Bibliogr. p. 132-134