Keele Research Repository

This thesis covers a number of related topics in the foundations of set theory. Section 1 contains an analysis of the paradoxes which suggests a way of looking at the axioms of any formalised set theory as to a certain extent legislating how 'is a member of' and 'set' are to be used.
In section 2, two other points of view are examined and rejected. There are doubtless other views which directly contradict the view put forward in section 1, but clearly one cannot examine all the different solutions of the paradoxes in any one thesis. The positions examined in section 2 seem to represent views that are most opposed to the general tendency of this thesis.
Section 3 consists of a critical examination of Tarski's
work on the semantic conception of truth for a formalised calculus of classes. This section has two aspects. The first is a defence of Tarski against some of his critics; the second an attack on certain of Tarski's conclusions.
Section 4 begins where section 3 leaves off.
it examines Tarski's conclusions about a definition of truth for set theory. It contains an analysis of Godel's results on the incompleteness of formal systems and is particularly concerned with the transference of certain inferences made from Godel's theorem for a formalised arithmetic to a formalised set theory. The conclusion of section 4 returns to the theme of section 1 and utilises certain metatheorems recently proved about formalised set theories.
Throughout the thesis no one axiom system of set theory is presupposed though reference will be made to several. The discussion is of a general nature and can be taken to be about any axiom system of set theory rather than some particular one.