Abstract

How intuitive are the Zermelo-Fraenkel axioms for set theory? This dissertation deals with this question by examining intuitive pictures and determining the extent to which each picture may guide us to accept the ZF axioms. ;Chapter 0 presents considerations of method, including a distinction between axioms that are intuitively required by a picture , and axioms that are rendered intuitively attractive . ;Chapter 1 explores the limitation-of-size picture, in which sets are portrayed as sufficiently small classes. This picture provides support for most of the ZF axioms, including the axiom of choice , but excluding the power-set and regularity axioms . Much of this intuitive support derives from a neglected idea of von Neumann, according to which every proper class is the same size as the universal class; this suggestion appears intuitively attractive in the size picture. ;Chapter 2 investigates the spatial picture, which arranges sets in a spatially organized hierarchy. This picture supports a version of the axioms due to Montague and Scott, as well as enough additional axioms to imply all the ZF axioms except replacement and AC. In the case of replacement, impredicativity is the obstacle to intuitive support. ;Chapter 3 examines the temporal picture, which, like Godel's iterative conception of sets, arranges sets in a temporally organized hierarchy. This picture supports all the axioms of the spatial picture together with mathematically significant instances of replacement. AC and impredicative instances of replacement remain problematic. ;The concluding remarks urge the adoption of a plurality of intuitive standpoints, which has the advantage of supporting all the ZF axioms. But, since no single picture supports every axiom, we must not regard ZF as an inevitable monolith