On the principle of dependent choices

Abstract

Publisher Summary This chapter discusses the general axiom of choice (denoted by (Z)) is independent of (T), that is, cannot be proved from (T) and the usual axioms of set-theory. An independence-proof has sense only with respect to a well-defined formal system whose consistency is either proved or assumed as an hypothesis. Proof applies only to such systems of set-theory that remain self-consistent after the adjunction of the following axiom: there is a non-denumerable set of elements which are not sets.