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Set Theory and its Philosophy

 Oxford University Press, 2004STAIP image

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.


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


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


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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'WV'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'VV'W. Hence W = {V' : V'V}. Contradiction. ☐