Adding Dependent Choice to the Prime Ideal Theorem

Abstract

This chapter presents a model of ZF set theory in which the principle of dependent choices and the prime ideal theorem for Boolean algebras are both true while the axiom of choice is false. The problem of finding such a model remained open despite considerable effort. It contains a model that is considered to be the best candidate for the theorem, together with an incomplete proof sketch along the lines of the original independence proof of the axiom of choice from the prime ideal theorem. No progress has been made on this model. The chapter presents some further applications of the method without proof. The prime ideal theorem, Hahn Banach theorem, and canonical uniform ultrafilter principle (a uniform ultrafilter includes all sets whose complements are well orderable and have smaller cardinality. The principle states that there is a function assigning a uniform ultrafilter to the power set of each infinite set) can be added to the class of automatic ZF transferable Fraenkel–Mostowski independences.