Subsystems of Second Order Arithmetic

449 Pages·1999·1.25 MB·English

by  Stephen G. Simpson

Perspectives
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In
Mathematical Logic
Editors
S. Feferman W. A. Hodges
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Stephen G. Simpson
Subsystems
of Second Order
Arithmetic
Stephen G. Simpson
Department of Mathematics
Pennsylvania State University
333 McAllister Building
University Park, PA 16802, USA
e-mail: [email protected]
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek -CIP-Eiuheitsaufnahme
Simpson, Stephen G.:
Subsystems of second order arithmetic / Stephen G. Simpson. -Berliu
; Heidelberg; New York; Barcelona; Budapest ; Hong Kong;
London; Milan; Singapore; Tokyo: Springer, 1999
(Perspectives in mathematical logic)
Mathematics Subject Classification (1991): 03B30,03F35
ISSN 0172-6641
ISBN-13: 978-3-642-64203-6 e-ISBN-13: 978-3-642-59971-2
DOl: 10.1007/978-3-642-59971-2
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Perspectives in Mathematical Logic
This series was founded in 1969 by the Omega Group consisting of R. O.
Gandy, H. Hermes, A. Levy, G. H. Muller, G. E. Sacks and D. S. Scott.
Initially sponsored by a grant from the Stiftung Volkswagenwerk, the
series appeared under the auspices of the Heidelberger Akademie der
Wissenschaften. Since 1986, Perspectives in Mathematical Logic is
published under the auspices of the Association for Symbolic Logic.
Mathematical Logic is a subject which is both rich and varied. Its origins
lie in philosophy and the foundations of mathematics. But during the
last half century it has formed deep links with algebra, geometry,
analysis and other branches of mathematics. More recently it has
become a central theme in theoretical computer science, and its influence
in linguistics is growing fast.
The books in the series differ in level. Some are introductory texts
suitable for final year undergraduate or first year graduate courses,
while others are specialized monographs. Some are expositions ofw ell
established material, some are at the frontiers of research. Each offers
an illuminating perspective for its intended audience.
Preface
Foundations of mathematics is the study of the most basic concepts and
logical structure of mathematics, with an eye to the unity of human knowl
edge. Among the most basic mathematical concepts are: number, shape, set,
function, algorithm, mathematical axiom, mathematical definition, mathe
matical proof. Typical questions in foundations of mathematics are: What
is a number? What is a shape? What is a set? What is a function? What
is an algorithm? What is a mathematical axiom? What is a mathematical
definition? What is a mathematical proof? What are the most basic concepts
of mathematics? What is the logical structure of mathematics? What are
the appropriate axioms for numbers? What are the appropriate axioms for
shapes? What are the appropriate axioms for sets? What are the appropriate
axioms for functions? Etc., etc.
Obviously foundations of mathematics is a subject which is of the great
est mathematical and philosophical importance. Beyond this, foundations of
mathematics is a rich subject with a long history, going back to Aristotle
and Euclid and continuing in the hands of outstanding modern figures such
as Descartes, Cauchy, WeierstraB, Dedekind, Peano, Frege, Russell, Cantor,
Hilbert, Brouwer, Weyl, von Neumann, Skolem, Tarski, Heyting, and G6del.
An excellent reference for the modern era in foundations of mathematics is
van Heijenoort [269].
In the late 19th and early 20th centuries, virtually all leading mathe
maticians were intensely interested in foundations of mathematics and spoke
and wrote extensively on this subject. Today that is no longer the case. Re
grettably, foundations of mathematics is now out of fashion. Today, most of
the leading mathematicians are ignorant of foundations and focus mostly on
structural questions. Today, foundations of mathematics is out of favor even
among mathematical logicians, the majority of whom prefer to concentrate
on methodological or other non-foundational issues.
This book is a contribution to foundations of mathematics. Almost all of
the problems studied in this book are motivated by an overriding founda
tional question: What are the appropriate axioms for mathematics? We un
dertake a series of case studies to discover which are the appropriate axioms
for proving particular theorems in core mathematical areas such as algebra,
analysis, and topology. We focus on the language of second order arithmetic,
VIII Preface
because that language is the weakest one that is rich enough to express and
develop the bulk of core mathematics. It turns out that, in many particu
lar cases, if a mathematical theorem is proved from appropriately weak set
existence axioms, then the axioms will be logically equivalent to the theo
rem. Furthermore, only a few specific set existence axioms arise repeatedly
in this context: recursive comprehension, weak Konig's lemma, arithmetical
IIt
comprehension, arithmetical transfinite recursion, comprehension; corre
sponding to the formal systems RCAo, WKLo, ACAo, ATRo, IIt-CAo; which
in turn correspond to classical foundational programs: constructivism, fini
tistic reductionism, predicativism, and predicative reductionism. This is the
theme of Reverse Mathematics, which dominates part A of this book. Part B
focuses on models of these and other subsystems of second order arithmetic.
Additional results are presented in an appendix.
The formalization of mathematics within second order arithmetic goes
back to Dedekind and was developed further by Hilbert and Bernays in
[114, supplement IV]. The present book may be viewed as a continuation
of Hilbert/Bernays [114]. I hope that the present book will help to revive the
study of foundations of mathematics and thereby earn for itself a permanent
place in the history of the subject.
Serious students of foundations of mathematics are invited to join in the
electronic discussion at www . math. psu. edu/ s impson/ f om/. The web page for
this book is www.math.psu.edu/simpson/sosoa/.
Acknowledgements
Much of my work on subsystems of second order arithmetic has been carried
on in collaboration with my students at Berkeley and Penn State, includ
ing: Stephen Brackin, Douglas Brown, Qi Feng, Fernando Ferreira, Mariag
nese Giusto, Kostas Hatzikiriakou, Jeffry Hirst, James Humphreys, Michael
Jamieson, Alberto Marcone, John Steel, Rick Smith, Robert Van Wesep,
Galen Weitkamp, Xiaokang Yu. I also acknowledge the collaboration and en
couragement of numerous colleagues including: Peter Aczel, Jeremy Av igad,
Jon Barwise, Michael Beeson, Errett Bishop, Andreas Blass, Lenore Blum,
Douglas Bridges, Wilfried Buchholz, Chi-Tat Chong, Rolando Chuaqui, Pe
ter Clote, Carlos Di Prisco, Rod Downey, Robin Gandy, Victor Harnik,
Leo Harrington, Petr Hajek, Ward Henson, Peter Hinman, William Howard,
Martin Hyland, Gerhard Jager, Haim Judah, Irving Kaplansky, Alexan
der Kechris, Jerome Keisler, Stephen Kleene, Julia Knight, Georg Kreisel,
Antonin Kucera, Richard Laver, Steffen Lempp, Manuel Lerman, Azriel
Levy, Alain Louveau, Angus Macintyre, Michael Makkai, Richard Mansfield,
David Marker, Donald Martin, Adrian Mathias, Kenneth McAloon, George
Metakides, Grigori Mints, Yiannis Moschovakis, Gert Miiller, Jan Mycielski,
Anil Nerode, Charles Parsons, Marian Pour-EI, Jean-Pierre Ressayre, Ian
Richards, Hartley Rogers, Gerald Sacks, Ramez Sami, Andre Scedrov, James
Schmerl, Kurt Schiitte, Helmut Schwichtenberg, Dana Scott, Wilfried Sieg,
Jack Silver, Saharon Shelah, John Shepherdson, Joseph Shoenfield, Richard
Shore, Theodore Slaman, Craig Smorynski, Robert Soare, Robert Solovay,
Rick Sommer, Gaisi Takeuti, Kazuyuki Tanaka, Dirk van Dalen, Lou van
den Dries, Daniel Velleman, Stan Wainer, Dongping Yang, and especially
Solomon Feferman, Harvey Friedman, Carl Jockusch, and Wolfram Pohlers.
I apologize to the many people whom I have disappointed by not finishing
this book sooner. Some of those individuals are mentioned above.
An early version of this book was written with a software package known
as MathText. I received important help from Robert Huff, the author of
MathText, and Janet Huff. Padma Raghavan wrote additional software to
help me convert the manuscript from MathText to LaTeX.
I acknowledge the help of various institutions including: the Alfred
P. Sloan Foundation, the American Mathematical Society, the Associa
tion for Symbolic Logic, the Centre National de Recherche Scientifique,
X Acknowledgements
the Deutsche Forschungsgemeinschaft, the National Science Foundation, the
Omega Group, Oxford University, the Pennsylvania State University, the
Raymond N. Shibley Foundation, the Science Research Council, Springer
Verlag, Stanford University, the University of California at Berkeley, the
University of TIlinois at Urbana/Champaign, the University of Munich, the
University of Paris, the University of Tennessee, the Volkswagen Foundation.
I thank my darling wife, Padma Raghavan, for her encouragement and
emotional support while I was bringing this project to a conclusion.
August 1998 Stephen G. Simpson
Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. VII
Acknowledgements ........................................... IX
Table of Contents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. XI
I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Main Question. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Subsystems of Z2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 The System ACAo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Mathematics Within ACAo .............................. 9
1.5 IIi-CAo and Stronger Systems . . . . . . . . . . . . . . . . . . . . . . . . . .. 15
1.6 Mathematics Within IIt-CAo '" . . . . . . . . . . . . . . . . . . . . . . . .. 18
1. 7 The System RCAo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23
1.8 Mathematics Within RCAo ..................... . . . . . . . .. 26
1.9 Reverse Mathematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31
1.10 The System WKLo ..................................... 35
1.11 The System ATRo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37
1.12 The Main Question, Revisited ........................... 41
1.13 Outline of Chapters II Through X . . . . . . . . . . . . . . . . . . . . . . .. 43
1.14 Conclusions ........................................... 59
Part A. Development of Mathematics
Within Subsystems of Z2
II. Recursive Comprehension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 63
II.l The Formal System RCAo .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 63
11.2 Finite Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 65
11.3 Primitive Recursion .................................... 69
II.4 The Number Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 73
11.5 Complete Separable Metric Spaces ....................... 78
II.6 Continuous Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 84
II.7 More on Complete Separable Metric Spaces. . . . . . . . . . . . . . .. 88