This book is designed for readers who know

elementary mathematical logic and axiomatic set

theory, and who want to learn more about set theory.

The primary focus of the book is on the independence

proofs. Most famous among these is the independence

of the Continuum Hypothesis (CH); that is, there are

models of the axioms of set theory (ZFC) in which

CH is true, and other models in which CH is false.

More generally, cardinal exponentiation on the regular

cardinals can consistently be anything not contradicting

the classical theorems of Cantor and König.

The basic methods for the independence proofs are

the notion of constructibility, introduced by Gödel, and

the method of forcing, introduced by Cohen. This book

describes these methods in detail, verifi es the basic

independence results for cardinal exponentiation, and

also applies these methods to prove the independence

of various mathematical questions in measure theory

and general topology.

Before the chapters on forcing, there is a fairly long

chapter on "infi nitary combinatorics". This consists

of just mathematical theorems (not independence

results), but it stresses the areas of mathematics

where set-theoretic topics (such as cardinal arithmetic)

are relevant.

There is, in fact, an interplay between infi nitary

combinatorics and independence proofs. Infi nitary

combinatorics suggests many set-theoretic questions

that turn out to be independent of ZFC, but it also

provides the basic tools used in forcing arguments. In

particular, Martin's Axiom, which is one of the topics

under infi nitary combinatorics, introduces many of the

basic ingredients of forcing.