Set Theory and Foundations of Mathematics
An Introduction to Mathematical Logic
Volume I
Set Theory
Set Theory and Foundations of Mathematics
An Introduction to Mathematical Logic
Volume I
Set Theory
Douglas Cenzer
University of Florida, USA
Jean Larson
University of Florida, USA
Christopher Porter
Drake University, USA
Jindrich Zapletal
University of Florida, USA
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Control Number: 2020007602
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
SET THEORY AND FOUNDATIONS OF MATHEMATICS: AN INTRODUCTION TO MATHEMATICAL LOGICVolume I: Set Theory
Copyright © 2020 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN 978-981-120-192-9 (hardcover)
ISBN 978-981-120-193-6 (ebook for institutions)
ISBN 978-981-120-194-3 (ebook for individuals)
For any available supplementary material, please visit
https://www.worldscientific.com/worldscibooks/10.1142/11324#t=suppl
Desk Editors: V. Vishnu Mohan/Kwong Lai Fun
Typeset by Stallion Press
Email: [email protected]
Printed in Singapore
Preface
This book was developed over many years from class notes for a set theory course at the University of Florida. This course has been taught to advanced undergraduates as well as lower level graduate students. The notes have been used more than 30 times as the course has evolved from seminar-style towards a more traditional lecture.
Axiomatic set theory, along with logic, provides the foundation for higher mathematics. This book is focused on the axioms and how they are used to develop the universe of sets, including the integers, rational and real numbers, and transfinite ordinal and cardinal numbers. There is an effort to connect set theory with the mathematics of the real numbers. There are details on various formulations and applications of the Axiom of Choice. Several special topics are covered. The rationals and the reals are studied as dense linear orderings without endpoints. The possible types of well-ordered subsets of the rationals and reals are examined. The possible cardinality of sets of reals is studied. The Cantor space 2 and Baire space
It is reasonable to cover most of the material in a one semester course, with selective omissions. Chapter 2 is a review of sets and logic, and should be covered as needed in one or two weeks. Chapter 3 introduces the Axioms of Zermelo–Fraenkel, as well as the Axiom of Choice, in about two weeks. Chapter 4 develops the Natural Numbers, induction and recursion, and introduces cardinality, taking two or three weeks. Chapter 5 on Ordinal Numbers includes transfinite induction and recursion, well-ordering, and ordinal arithmetic, in two or three weeks. Chapter 6 covers equivalent versions and applications of the Axiom of Choice, as well as Cardinality, in about two or three weeks. The Real Numbers are developed in Chapter 7, with discussion of dense and complete orders, countable and uncountable sets of reals, and a brief introduction to topological spaces such as the Baire space and Cantor space, again in two or three weeks. If all goes well, this leaves about one week each for the final two chapters: Models of Set Theory and an introduction to Ramsey theory.
The book contains nearly 300 exercises which test the students understanding and also enhance the material.
The authors have enjoyed teaching from these notes and are very pleased to share them with a broader audience.
About the Authors
Douglas Cenzer is Professor of Mathematics at the University of Florida, where he was Department Chair from 2013 to 2018. He has to his credit more than 100 research publications, specializing in computability, complexity, and randomness. He joined the University of Florida in 1972 after receiving his Ph.D. in mathematics from the University of Michigan.
Jean Larson is Emeritus Professor of Mathematics at the University of Florida, specializing in combinatorial set theory. She received her Ph.D. in mathematics from Dartmouth University in 1972 and was E.R. Hedrick Assistant Professor at UCLA from 1972 to 1974, before joining the University of Florida in 1974.
Christopher Porter is Assistant Professor of Mathematics at Drake University, specializing in computability theory, algorithmic randomness, and the philosophy of mathematics. He received his Ph.D. in mathematics and philosophy from the University of Notre Dame in 2012, was an NSF international postdoctoral fellow at Université Paris