The Axioms of Set Theory.pdf - Department of Pure Mathematics and ...

The Axioms of Set Theory Thomas Forster

Contents

1

Preface

page 4

2

The Cumulative Hierarchy

3 3.1 3.2 3.3 3.4

Some Philosophical Prolegomena Inference to the best explanation Intension and Extension What is a Mathematical Object? The Worries about Circularity

6 8 8 9 10 13

4 Some History 4.1 What are sets anyway? 4.1.1 Set Pictures

16 18 23

5 5.1 5.2 5.3 5.4 5.5 5.5.1

Listing the Axioms First Bundle: The Axiom of Extensionality Second Bundle: The Closure Axioms Third Bundle: The Axioms of infinity Fourth Bundle The Axiom of Foundation The Remaining Axioms

25 25 26 27 28 29 33

6 6.1 6.1.1 6.1.2 6.1.3 6.1.4 6.2 6.3 6.3.1

Replacement and Collection Limitation of Size Church’s distinction between high and intermediate sets LOS and some proofs Foundation and Replacement Replacement not consistent with limitation of size? Is Replacement just true? Reasons for adopting the axiom Facts about Vω+ω

34 35 36 38 38 39 40 41 41

3

4

0 Contents

6.3.2 G¨ odel’s Argument 6.3.3 The Argument from the Normal Form Theorem for Restricted Quantifiers 6.3.4 The Argument from Implementation-invariance 6.3.5 Existence of Inductively defined sets 6.3.6 Existence of Transitive Closures

44 46 50 53

7 7.1 7.1.1 7.2 7.2.1 7.2.2 7.2.3 7.3 7.3.1 7.3.2 7.3.3 7.4 7.5 7.5.1

The Axiom of Choice IBE and some counterexamples Constructive Mathematicians do not like AC IBE and a Fallacy of Equivocation Socks A union of countably many countable sets is countable Every perfect binary tree has an infinite path The Fallacy of Equivocation Perfect binary trees The Countable Union of Countable Sets Socks AC keeps thing simple Is AC true? The Consistency of the Axiom of Choice?

57 61 62 64 64 65 66 67 67 68 68 70 71 74

8 8.1 8.2 8.3 8.4 8.5 8.5.1 8.6 8.6.1 8.7 8.8

Independence Proofs Replacement Power set Infinity Sumset Foundation Antifoundation Extensionality More about Extensionality Choice Pairing

75 78 78 79 79 80 80 81 82 82 84

9 ZF with Classes 9.0.1 Global Choice 9.0.2 Von Neumann’s axiom Glossary Bibliography

43

86 88 88 90 94

1 Preface

This is not intended to be an introductory text in set theory: there are plenty of those already. It’s designed to do exactly what it says on the tin: to introduce the reader to the axioms of Set Theory. And by ‘Set theory’ here I mean the axioms of the usual system of ZermeloFraenkel set theory, including at least some of the fancy add-ons that do not come as standard.1 Its intention is to explain what the axioms say, why we might want to adopt them (in the light of the uses to which they can be put) say a bit (but only a bit, for this is not a historical document) on how we came to adopt them, and explain their mutual independence. Among the things it does not set out to do is develop set theory axiomatically: such deductions as are here drawn out from the axioms are performed solely in the course of an explanation of why an axiom came to be adopted; it contains no defence of the axiomatic method; nor is it a book on the history of set theory. I am no historian, and the historical details of the debates attending their adoption and who did what and with which and to whom are of concern to me only to the extent that they might help me in the task explaining what the axioms say and why one might want to adopt them. Finally I must cover myself by pointing out in my defence that I am not an advocate for any foundational rˆole for set theory: it is a sufficient justification for a little book like this merely that there are a lot of people who think that set theory has a foundational rˆole: it’s a worthwhile exercise even if they are wrong. Other essays with a brief like the one I have given myself here include 1

There are other systems of axioms, like those of Quine’s New Foundations, Church’s set theory CUS, and the Positive Set Theory studied by the School around Roland Hinnion at the Universit´ e Libre de Bruxelles, but we will mention them only to the extent that they can shed light on the mainstream material.

5

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1 Preface

Mycielski [35] and Shoenfield [44]. My effort is both more elementary and more general than theirs are.2 Whom is it for? Various people might be interested. People in Theoretical Computer Science, mathematicians, and the gradually growing band of people in Philosophy who are developing an interest in Philosophy of Mathematics all come to mind. However one result of my attempts to address simultaneously the concerns of these different communities (as I discover from referees’ reports) is that every time I put in a silver threepenny bit for one of them one of the others complains that they have cracked their teeth on it. This document was prepared in the first instance for my set theory students at Cambridge, so it should come as no surprise that the background it relies on can be found in a home-grown text: [17]. The fact that [17] is an undergraduate text should calm the fears of readers concerned that they might not be getting a sufficiently elementary treatment. It is a pleasure to be able to thank Ben Garling, Akihiro Kanamori, Adrian Mathias, Robert Black, Douglas Bridges, Imre Leader, Nathan Bowler, Graham White, Allen Hazen (and others, including some anonymous referees) for useful advice, and thanks to my students for invaluable feedback.

2

Despite the promising-sounding title Lemmon [27] is a technical work.

2 The Cumulative Hierarchy

The axioms of set theory of my title are the axioms of Zermelo-Fraenkel set theory, usually thought of as arising from the endeavour to axiomatise the cumulative hierarchy concept of set. There are other conceptions of set, but although they have genuine mathematical interest they are not our concern here. The cumulative hierarchy of sets is built in an arena—which is initially empty—of sets, to which new sets are added by a process (evocatively called lassoing by Kripke) of making new sets from collections of old, pre¨existing sets. No set is ever harmed in the process of making new sets from old, so the sets accumulate: hence ‘cumulative’. Formally we can write [ P(Vβ ) (2.1) Vα =: β