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.

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