This simple introductory treatment emphasizes fundamentals and is easily
comprehensible to readers with some college algebra. Topics include
rudiments; arbitrary sets and their cardinal numbers; ordered sets and
their ordered types; and well-ordered sets and their ordinal numbers.
"Exceptionally well written." — School Science and Mathematics. "A very fine book."
A century ago, Georg Cantor demonstrated the possibility of a series of
transfinite infinite numbers. His methods, unorthodox for the time,
enabled him to derive theorems that established a mathematical reality
for a hierarchy of infinities. Cantor's innovation was opposed, and
ignored, by the establishment; years later, the value of his work was
recognized and appreciated as a landmark in mathematical thought,
forming the beginning of set theory and the foundation for most of
contemporary mathematics.
Studies in Logic and the Foundations of Mathematics, Volume 102: Set
Theory: An Introduction to Independence Proofs offers an introduction to
relative consistency proofs in axiomatic set theory, including
combinatorics, sets, trees, and forcing.
Set theory is the mathematics of infinity and part of the core
curriculum for mathematics majors. This book blends theory and
connections with other parts of mathematics so that readers can
understand the place of set theory within the wider context. Beginning
with the theoretical fundamentals, the author proceeds to illustrate
applications to topology, analysis and combinatorics, as well as to pure
set theory. Concepts such as Boolean algebras, trees, games, dense
linear orderings, ideals, filters and club and stationary sets are also
developed. Pitched specifically at undergraduate students, the approach
is neither esoteric nor encyclopedic. The author, an experienced
instructor, includes motivating examples and over 100 exercises designed
for homework assignments, reviews and exams. It is appropriate for
undergraduates as a course textbook or for self-study. Graduate students
and researchers will also find it useful as a refresher or to solidify
their understanding of basic set theory.
This exploration of a notorious mathematical problem is the work of the
man who discovered the solution. The independence of the continuum
hypothesis is the focus of this study by Paul J. Cohen. It presents not
only an accessible technical explanation of the author's landmark proof
but also a fine introduction to mathematical logic. An emeritus
professor of mathematics at Stanford University, Dr. Cohen won two of
the most prestigious awards in mathematics: in 1964, he was awarded the
American Mathematical Society's Bôcher Prize for analysis; and in 1966,
he received the Fields Medal for Logic.
This text covers the parts of contemporary set theory relevant to other
areas of pure mathematics. After a review of "naïve" set theory, it
develops the Zermelo-Fraenkel axioms of the theory before discussing the
ordinal and cardinal numbers. It then delves into contemporary set
theory, covering such topics as the Borel hierarchy and Lebesgue
measure. A final chapter presents an alternative conception of set
theory useful in computer science.
This clear and well-developed approach to axiomatic set theory is geared
toward upper-level undergraduates and graduate students. It
examines the basic paradoxes and history of set theory and advanced
topics such as relations and functions, equipollence, finite sets and
cardinal numbers, rational and real numbers, and other subjects.
This is an introductory undergraduate textbook in set theory. In
mathematics these days, essentially everything is a set. Some knowledge
of set theory is necessary part of the background everyone needs for
further study of mathematics. It is also possible to study set theory
for its own interest--it is a subject with intruiging results anout
simple objects. This book starts with material that nobody can do
without. There is no end to what can be learned of set theory, but here
is a beginning.
Set Theory and Logic is the result of a course of lectures for
advanced undergraduates, developed at Oberlin College for the purpose of
introducing students to the conceptual foundations of mathematics.
Mathematics, specifically the real number system, is approached as a
unity whose operations can be logically ordered through axioms. One of
the most complex and essential of modern mathematical innovations, the
theory of sets (crucial to quantum mechanics and other sciences), is
introduced in a most careful concept manner, aiming for the maximum in
clarity and stimulation for further study in set logic.
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.
Set theory permeates much of contemporary mathematical thought. This
text for undergraduates offers a natural introduction, developing the
subject through observations of the physical world. Its progressive
development leads from concrete finite sets to cardinal numbers,
infinite cardinals, and ordinals.
2011 Reprint of 1960 Edition. Full facsimile of the original edition,
not reproduced with Optical Recognition Software. Paul Richard Halmos
(1916-2006) was a Hungarian-born American mathematician who made
fundamental advances in the areas of probability theory, statistics,
operator theory, ergodic theory, and functional analysis (in particular,
Hilbert spaces). He was also recognized as a great mathematical
expositor. "...He (the author) uses the language and notation of
ordinary informal mathematics to state the basic set-theoretic facts
which a beginning student of advanced mathematics needs to
know...Because of the informal method of presentation, the book is
eminently suited for use as a textbook or for self-study. The reader
should derive from this volume a maximum of understanding of the
theorems of set theory and of their basic importance in the study of
mathematics." - "Philosophy and Phenomenological Research".
Suitable for upper-level undergraduates, this accessible approach to set
theory poses rigorous but simple arguments. Each definition is
accompanied by commentary that motivates and explains new concepts.
Starting with a repetition of the familiar arguments of elementary set
theory, the level of abstract thinking gradually rises for a progressive
increase in complexity.
Designed for undergraduate students of set theory, Classic Set Theory
presents a modern perspective of the classic work of Georg Cantor and
Richard Dedekin and their immediate successors. This includes:
This work is a translation into English of the Third Edition of the classic German language work Mengenlehre by Felix Hausdorff published in 1937. From the Preface (1937): ``The present book has as its purpose an exposition of the most important theorems of the theory of sets, along with complete proofs, so that the reader should not find it necessary to go outside this book for supplementary details while, on the other hand, the book should enable him to undertake a more detailed study of the voluminous literature on the subject. The book does not presuppose any mathematical knowledge beyond the differential and integral calculus, but it does require a certain maturity in abstract reasoning; qualified college seniors and first year graduate students should have no difficulty in making the material their own ... The mathematician will ... find in this book some things that will be new to him, at least as regards formal presentation and, in particular, as regards the strengthening of theorems, the simplification of proofs, and the removal of unnecessary hypotheses.''