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Building a theory of sets

The axiomatic development of set theory is among the most impressive accomplishments of modern logic. It can be used to give precise meaning to concepts which were beyond the grasp of its vague predecessors. A successful set theory describes clearly the logical and extra-logical principles of mathematics.

We want a theory of sets to be at least powerful enough to cope with the concepts of classical mathematics, in particular we need to be able to speak about the number systems discussed in the introduction. We have seen that the systems of integers, rational numbers, real numbers, complex numbers and algebraic numbers (and beyond) can be built from the natural numbers using a handful of logical constructions. Thus our theory needs to be capable of describing a model of the natural numbers, that is, a collection of sets with a successor operator satisfying Peano's Postulates, together with such notions as ordered pairs, functions and other relations of various kinds. At the same time, and this is where the creative tension comes into play, we don't want the theory to be so loose and overconfident with its assignment of sets to admit such horrors as Russell's paradox.