ARITHMETICAL KNOWLEDGE AND ARITHMETICAL DEFINABILITY

334 Pages·2010·2.39 MB·English

ARITHMETICAL KNOWLEDGE AND ARITHMETICAL DEFINABILITY:
FOUR STUDIES
A Dissertation
Submitted to the Graduate School
of the University of Notre Dame
in Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy
by
Sean Walsh,
Michael Detlefsen, Co-Director
Peter Cholak, Co-Director
Graduate Programs in Philosophy and Mathematics
Notre Dame, Indiana
December 2010
(cid:13)c Copyright by
Sean Walsh
2010
All Rights Reserved
ARITHMETICAL KNOWLEDGE AND ARITHMETICAL DEFINABILITY:
FOUR STUDIES
Abstract
by
Sean Walsh
The subject of this dissertation is arithmetical knowledge and arithmetical
definability. The first two chapters contain respectively a critique of a logicist
account of a preferred means by which we may legitimately infer to arithmetical
truths and a tentative defense of an empiricist account. According to the logicist
account, one may infer from quasi-logical truths to patently arithmetical truths
because the arithmetical truths are representable in the logical truths. It is argued
in the first chapter that this account is subject to various problems: for instance,
the most straightforward versions seem vulnerable to various counterexamples.
The basic idea of the alternative empiricist account considered in chapter two is
that complicated arithmetical truths like mathematical induction may be inferred
by way of confirmation from less complicated quantifier-free arithmetical truths.
The notion of confirmation here is understood probabilistically, and responses
are given in this chapter to several seeming problems with this importation of
probability into arithmetic.
The final two chapters are concerned with arithmetical definability in two
different settings. In the third chapter, the interpretability strength of the arith-
metical and hyperarithmetical subsystems of second-order Peano arithmetic is
Sean Walsh
compared to the interpretability strength of analogous systems centered around
twoprinciplescalledHume’sPrincipleandBasicLawV,whichrespectivelyaxiom-
atize a standard notion of cardinality and an alternative conception of set. One of
the major results of this chapter is that the hyperarithmetic subsystem of Hume’s
Principle does not interpret the hyperarithmetic subsystem of second-order Peano
arithmetic. The fourth chapter is concerned with arithmetical definability in the
setting of descriptive set theory, where the relevant benchmark is between notions
which may be defined without quantification over elements of certain topological
spaces (Borel notions) and notions whose definitions do require such quantifica-
tion (analytic, coanalytic, projective notions). In this fourth chapter the Denjoy
integral is studied from the vantage point of descriptive set theory, and it is shown
that the graph of the indefinite integral is not Borel but rather is properly coana-
lytic. This contrasts to the Lebesgue integral, which is Borel under this measure
of complexity.
To my father, for always encouraging me to see where the circles cross.
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CONTENTS
FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
CHAPTER 1: LOGICISM, INTERPRETABILITY, AND KNOWLEDGE
OF ARITHMETIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction: The Logicist Template . . . . . . . . . . . . . . . . 1
1.2 Background: The Interpretability of Theories and Structures . . . 6
1.3 Theory-Based Versions: the Plethora and Consistency Problems . 12
1.4 Structure-Based Version: the Isomorphism and Signature Problems 24
1.5 Conclusions and Directions for Further Research . . . . . . . . . . 40
1.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
CHAPTER 2: EMPIRICISM, PROBABILITY, AND KNOWLEDGE OF
ARITHMETIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.1 Introduction: Inceptive and Amplificatory Empiricism . . . . . . . 59
2.2 Challenges to Access to Probability Assignments . . . . . . . . . . 67
2.2.1 Countable Additivity: Aligning the True and Probable . . 68
2.2.2 The Non-Computability of Probability Assignments . . . . 77
2.3 Challenges to Arithmetical Instance Confirmation . . . . . . . . . 84
2.3.1 Baker and the Exigencies of Arithmetical Sampling . . . . 85
2.3.2 Stable and Unstable Reasoning in Geometry and Arithmetic 95
2.4 Challenges from Alternative Inferences . . . . . . . . . . . . . . . 106
2.5 Conclusions and Directions for Future Research . . . . . . . . . . 116
2.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
CHAPTER 3: COMPARING PEANO ARITHMETIC, BASIC LAW V,
AND HUME’S PRINCIPLE . . . . . . . . . . . . . . . . . . . . . . . . 149
3.1 Introduction, Definitions, and Overview of Main Results . . . . . 149
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3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 149
3.1.2 Definition of Signatures and Theories of PA2, BL2 and HP2 . 150
3.1.3 Definition of Subsystems of PA2, BL2 and HP2 . . . . . . . . 156
3.1.4 Summary of Results about the Provability Relation . . . . 160
3.1.5 Summary of Results about the Interpretability Relation . . 162
3.2 Standard Models of HP2 and Associated Results . . . . . . . . . . 166
3.2.1 Models of HP2 from Infinite Cardinals . . . . . . . . . . . . 167
3.2.2 The Mutual Interpretability of PA2 and HP2 . . . . . . . . . 175
3.3 Standard Models of Subsystems of BL2 and Associated Results . . 186
3.3.1 Generalities on Models of Subsystems of BL2 . . . . . . . . 186
3.3.2 Hyperarithmetic Theory and Related Results . . . . . . . 190
3.3.3 Standard Models of the Hyperarithmetic Subsystems of BL2 197
3.4 Barwise-Schlipf Models of Subsystems of BL2 and HP2 . . . . . . . 203
3.4.1 Generalized Barwise-Schlipf/Ferreira-Wehmeier Theorem . 203
3.4.2 Application to Algebraically Closed Fields . . . . . . . . . 212
3.4.3 Application to O-Minimal Expansions of Real-Closed Fields 220
3.4.4 Application to Separably Closed Fields . . . . . . . . . . . 227
3.5 Further Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
CHAPTER 4: DENJOY INTEGRATION: DESCRIPTIVE SET THEORY
AND MODEL THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . 237
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
4.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
4.2.1 Absolutely Continuous Functions and Generalizations . . . 240
4.2.2 Basic Properties of the Denjoy Integral . . . . . . . . . . . 246
4.2.3 Lebesgue’s Lemma and the Subspaces . . . . . . . . . . . . 250
4.3 Descriptive Set Theory . . . . . . . . . . . . . . . . . . . . . . . . 256
4.3.1 Three Derivatives and Functions of Arbitrarily High Rank 257
4.3.2 Totalization: Calibrating Rank and Entry into Subspaces . 266
4.3.3 Definability: The Derivatives are Borel . . . . . . . . . . . 274
4.4 Model Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
4.4.1 Indexes of Subgroups and Non-Definability of the Integral 289
4.4.2 Elementary Equivalence and Decidability . . . . . . . . . . 294
4.5 Further Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
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FIGURES
1.1 Summary of Problems for Versions of the Logicist Template . . . 42
2.1 Alternative Confirming Inferences: Two Pairs of Contrasting Infer-
ences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.1 Provability Relation in Subsystems of BL2, PA2, and HP2 . . . . . . 160
3.2 Interpretability Relation in Subsystems of BL2, PA2, and HP2 . . . 164
4.1 Containment Diagram for Subsets of M[a,b] and C[a,b] . . . . . . 241
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ACKNOWLEDGMENTS
I would first and foremost like to thank my wife Kari for her love and patience–
I wouldn’t have been able to do any of this without her. I would also like to
thank my parents, Kevin and Linda, and my in-laws, Ron and Annette, for their
persistent support and encouragement.
My advisors Michael Detlefsen and Peter Cholak have obviously shaped this
dissertation and my overall thinking in ways that I cannot begin to describe,
and I would like to thank them for their patience and help on my dissertation,
as well as for their continued support for my doing interdisciplinary work. I
would also like to thank my teachers at Notre Dame, and in particular Timothy
Bays, Patricia Blanchette, Curtis Franks, Julia Knight, and Sergei Starchenko,
for introducing me to the philosophy of mathematics and mathematical logic.
Likewise, my education at Notre Dame has been enriched by the many visitors
to, as well as former members of, the logic group at Notre Dame, including Pe-
ter Gerdes, Karen Lange, David Lippel, Colin McLarty, Serge Randriambololona,
Reed Solomon, Vitezslav Svejdar, and William Tait. Finally, I would like to thank
two of my teachers from Gonzaga University, Wayne Pomerleau and John Burke,
who first introduced me to so much of what I have come to love about philosophy
and mathematics.
During graduate school I have been the beneficiary of generous financial sup-
port from many institutions and groups, including the Philosophy Department at
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Notre Dame, the Mathematics Departments at Notre Dame, the Ahtna Heritage
Foundation, the Deutscher Akademischer Austausch Dienst, the George-August
Universita¨t Go¨ttingen, the National Science Foundation (under NSF Grants 02-
45167,EMSW21-RTG-03-53748,EMSW21-RTG-0739007,andDMS-0800198),the
Alexander von Humboldt Stiftung TransCoop Program, and the Ideals of Proof
Project, which in turn was funded and supported by Agence Nationale de la
Recherche, Universit´e Paris Diderot – Paris 7, Universit´e Nancy 2, Coll`ege de
France, and Notre Dame. There are of course many people behind these vari-
ous institutions and groups, and I would like to especially thank Karine Chemla,
BriceHalimi,GerhardHeinzmann,FelixMu¨hlho¨lzer,MarcoPanza,DavidRabouin,
Ivahn Smadja, Jean-Jacques Szczeciniarz, and Christian Tapp.
I have been working on the material collected here for several years, and have
benefited both from many opportunities to present components of this material at
various conferences and seminars, as well as opportunities to discuss this material
with my teachers and friends, and I would like to record some of these debts here.
A very early version of some of the underlying thoughts from Chapter 1 was
presented at the Sixth Annual Midwest Philosophy of Mathematics Workshop
held at Notre Dame on October 8, 2005 under the title “Justifications of Hume’s
Principle and Mathematical Induction.” In that talk, I was focused on under-
standing the history of various attempts to justify Hume’s Principle and math-
ematical induction, but in giving the talk I was forced to try to articulate the
epistemic relationship between these two principles, and it was that first attempt
that prompted the reflections currently found in Chapter 1. A very succinct ver-
sion of the material from Chapter 1 was presented in the first half of my talk
´
at Dr. Detlefsen’s Ideals of Proof Fellows’ Workshop held at the Ecole normale
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