Set Theory and its Philosophy 论文

摘要

Abstract This book presents a philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. The book offers an account of cardinal and ordinal arithmetic, and the various axiom candidates. It discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. The book offers a simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. The book interweaves a presentation of the technical material with a philosophical critique. The book does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true.