Set Theory

This book is designed for readers who knowelementary mathematical logic and axiomatic settheory, and who want to learn more about set theory.The primary focus of the book is on the independenceproofs. Most famous among these is the independenceof the Continuum Hypothesis (CH); that is, there aremodels of the axioms of set theory (ZFC) in whichCH is true, and other models in which CH is false.More generally, cardinal exponentiation on the regularcardinals can consistently be anything not contradictingthe classical theorems of Cantor and König.The basic methods for the independence proofs arethe notion of constructibility, introduced by Gödel, andthe method of forcing, introduced by Cohen. This bookdescribes these methods in detail, verifi es the basicindependence results for cardinal exponentiation, andalso applies these methods to prove the independenceof various mathematical questions in measure theoryand general topology.Before the chapters on forcing, there is a fairly longchapter on 'infi nitary combinatorics'. This consistsof just mathematical theorems (not independenceresults), but it stresses the areas of mathematicswhere set-theoretic topics (such as cardinal arithmetic)are relevant.There is, in fact, an interplay between infi nitarycombinatorics and independence proofs. Infi nitarycombinatorics suggests many set-theoretic questionsthat turn out to be independent of ZFC, but it alsoprovides the basic tools used in forcing arguments. Inparticular, Martin’s Axiom, which is one of the topicsunder infi nitary combinatorics, introduces many of thebasic ingredients of forcing.