What’s Wrong with Set Theory?

Abstract

I have posed the fundamental question in the philosophy of mathematics as ‘what does mathematics mean and how can we know for sure that it is true?’ Before developing my answer to this question, I must explain why I do not accept the orthodox answer, namely that mathematics is the study of sets and that our knowledge of mathematics is derived from our set-theoretic intuition using classical logic. In this chapter I shall argue that ‘set-theoretic intuition’, as formalised in the Zermelo-Fraenkel axioms with the axiom of choice (ZFC), is conceptually incoherent. In the following chapter I shall argue that infinite quantifiers, the distinctive feature of classical logic, are meaningless.