Fendrich, Samuel
(1987)
From axiomatization to generalizatrion of set theory.
PhD thesis, London School of Economics and Political Science.
Abstract
The thesis examines the philosophical and foundational significance of Cohen's Independence results. A distinction is made between the mathematical and logical analyses of the "set" concept. It is argued that topos theory is the natural generalization of the mathematical theory of sets and is the appropriate foundational response to the problems raised by Cohen's results. The thesis is divided into three parts. The first is a discussion of the relationship between "informal" mathematical theories and their formal axiomatic realizations this relationship being singularly problematic in the case of set theory. The second part deals with the development of the set concept within the mathemtical approach. In particular Skolem's reformulation of Zermlelo's notion of "definite properties". In the third part an account is given of the emergence and development of topos theory. Then the considerations of the first
two parts are applied to demonstrate that the shift to topos theory, specifically in its guise of LST (local set theory), is the appropriate next step in the evolution of the concept of set, within
the mathematical approach, in the light of the significance of Cohen's Independence results.
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