Abstract

Certain classical paradoxes of naive set theory are well known to arise through use of a naive comprehension principle. Later set theories, like the Zermelo-Fraenkel system , use only certain instances of comprehension and thereby avoid these classical paradoxes. There persists however the question of the consistency of these set theories. Being unable to prove their consistency from within, some heuristic argument for their consistency is sought. In the case of ZF a heuristic of "smallness" has been invoked: the paradoxes of naive set theory follow from incorrectly assuming certain large collections to be individuals ; ZF avoids the paradoxes because it requires that only certain small collections are individuals. This heuristic is known as the limitation of size theory. ;This thesis explicates the limitation of size theory by making precise several senses of small according to which the sets required by ZF are small. ;The simple theory of types is taken as the background theory with only six types necessary to the discussion. It is argued that the small type 1 collections form an ideal. The limitation of size idea that it is consistent to assume that small collections are individuals is then made precise in the assumption that there exists a 1-1 mapping from the ideal of small collections into the universe of individuals; that is, that small collections can be represented as individuals. ;Being merely an ideal is not enough to ensure the image of the collection of small collections is, with an appropriately defined membership relation, a model of all of ZF. Further conditions must be placed on the ideal to ensure that this is so. A generalization of finiteness, Tarski's notion of 'fixed by all permutations of the universe', and Cantor's 'subject to mathematical determination' are all identified as notions of smallness from which such conditions on ideals are derived. It is shown that the image of an ideal of collections small in these senses is a model of ZF