Set Theory and its Philosophy

 Oxford University Press, 2004

Hardback £78 ISBN 0-19-926973-4

Paperback £25 ISBN 0-19-927041-4

A comprehensive philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. I offer a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. I discuss in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. I offer a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unusual is that it interweaves a careful presentation of the technical material with a detailed philosophical critique: I do not merely expound the theory dogmatically but at every stage discuss in detail the reasons that can be offered for believing it to be true.

Contents

Part I: Sets

1. Logic
2. Collections
3. The hierarchy
4. The theory of sets

Part II: Numbers

1. Arithmetic
2. Counting
3. Lines
4. Real numbers

Part III: Cardinals and Ordinals

1. Cardinals
2. Basic cardinal arithmetic
3. Ordinals
4. Ordinal arithmetic

Part IV: Further Axioms

1. Orders of infinity
2. The axiom of choice
3. Further cardinal arithmetic

Appendices

1. Traditional axiomatizations
2. Classes
3. Sets and classes

Reviews

• Timothy Bays, Notre Dame Philosophical Reviews (2005)
• Stewart Shapiro, Mind 114 (2005) 764-7
• Arno Aurélio Viero, Manuscrito – Rev. Int. Fil. 28 (2005) 169-72
• Gabriel Uzquiano, Philosophia Mathematica 13 (2005) 308-46
• Elliott Mendelson, Zentralblatt 1066.03002
• Perry Smith, MR 2005d:03005

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Errata

p.44 l.14 For V ∪ V' read V ∩ V'.

p. 46 For the proof of proposition 3.6.13 substitute the following (thanks to Franz Fritsche):

Suppose that the proposition is false and V is the lowest level for which it fails, so that there is a history W of V such that W ≠ {V' : V' ∈ V}. Certainly V' ∈ W ⇒ V' ∈ V. So suppose now that V' ∉ W. Then for every V'' ∈ W we have V'' ≠ V' and V' ∉ V'' (since if V' ∈ V'' ∈ W then V' ∈ W), and so V'' ∈ V' [proposition 3.6.11]. So W ⊆ {V'' : V'' ∈ V'}, whence V = acc(W) ⊆ acc{V'' : V'' ∈ V'} = V' and therefore V' ∉ V (since otherwise V' ∈ V'). Contraposition gives V' ∈ V ⇒ V' ∈ W. Hence W = {V' : V' ∈ V}. Contradiction. ☐