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Open Set Theory OPEN SET THEORY Tim Button and the Open Logic Project About This book is an Open Education Resource. It is written for students with a little background in logic, and some high school mathematics. It aims to scratch the tip of the surface of the philosophy of set theory. By the end of this book, students reading it might have a sense of: 1. why set theory came about; 2. how to reduce large swathes of mathematics to set theory + arithmetic; 3. how to embed arithmetic in set theory; 4. what the cumulative iterative conception of set amounts to; 5. how one might try to justify the axioms of ZFC. The book grew out of a short course that I (Tim Button) teach at Cambridge. Before me, it was lectured by Luca Incurvati and Michael Potter. In writing this book—and the course, more generally—I was hugely indebted to both Luca and Michael. I want to offer both of them my heartfelt thanks. The book also draws material from the Open Logic Text. In particular, chap- ters1–3 are drawn (with only tiny changes) from the Open Logic Text. Recip- rocally, I (Tim) am contributing the material I have written back to the Open Logic Project. Please see openlogicproject.org for more information. This book was released January 21, 2019, under a Creative Commons license (CC BY 4.0; full details are available at the CC website). The following is a human-readable summary of (and not a substitute for) that license: License details. You are free to: — Share: copy and redistribute the material in any medium or format — Adapt: remix, transform, and build upon the material for any purpose, even commercially. Under the following terms: — Attribution: You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. 1 Contents 0 Prelude: history and mythology6 0.1 Infinitesimals and differentiation................. 6 0.2 Rigorous definition of limits ................... 8 0.3 Pathologies............................. 9 0.4 More myth than history? ..................... 11 0.5 Where we are headed....................... 11 I Na¨ıveset theory 13 1 Getting started 15 1.1 Extensionality ........................... 15 1.2 Subsets ............................... 16 1.3 Some important sets of numbers................. 17 1.4 Unions and Intersections ..................... 17 1.5 Pairs, Tuples, Cartesian Products................. 20 1.6 Russell’s Paradox ......................... 21 Problems.................................. 22 2 Relations 23 2.1 Relations as Sets .......................... 23 2.2 Philosophical reflections...................... 24 2.3 Special Properties of Relations.................. 26 2.4 Orders................................ 27 2.5 Operations on Relations...................... 29 Problems.................................. 30 3 Functions 31 3.1 Basics ................................ 31 3.2 Kinds of Functions......................... 33 3.3 Functions as relations....................... 35 3.4 Inverses of Functions ....................... 36 3.5 Composition of Functions..................... 37 Problems.................................. 37 2 CONTENTS 3 4 The Size of Sets 39 4.1 Introduction ............................ 39 4.2 Countable Sets........................... 39 4.3 Cantor’s zig-zag method ..................... 41 4.4 Pairing functions and codes.................... 42 4.5 Uncountable Sets.......................... 43 4.6 Reduction.............................. 45 4.7 Equinumerosity .......................... 45 4.8 Sets of different sizes, and Cantor’s Theorem.......... 46 4.9 The notion of size, and Schroder–Bernstein¨ ........... 47 4.10 Appendix: Cantor on the line and the plane.......... 48 4.11 Appendix: Hilbert’s space-filling curves ............ 49 Problems.................................. 51 5 Arithmetisation 53 5.1 From N to Z ............................ 53 5.2 From Z to Q ............................ 55 5.3 The real line............................. 56 5.4 From Q to R ............................ 57 5.5 Some philosophical reflections.................. 59 5.6 Appendix: ordered rings and fields............... 60 5.7 Appendix: the reals as Cauchy sequences............ 63 Problems.................................. 66 6 Infinite sets 68 6.1 Hilbert’s Hotel........................... 68 6.2 Dedekind algebras......................... 69 6.3 Dedekind algebras and arithmetical induction......... 70 6.4 Dedekind’s “proof” of an infinite set .............. 71 6.5 Appendix: a proof of Schroder–Bernstein¨ ............ 73 II The iterative conception 75 7 The iterative conception 78 7.1 Extensionality ........................... 78 7.2 Russell’s Paradox (again)..................... 78 7.3 Predicative and impredicative.................. 79 7.4 The cumulative-iterative approach................ 81 7.5 Urelements or not?......................... 82 7.6 Appendix: Frege’s Basic Law V ................. 84 8 Steps towards Z 85 8.1 The story in more detail...................... 85 8.2 Separation.............................. 85 8.3 Union ................................ 87 CONTENTS 4 8.4 Pairs................................. 87 8.5 Powersets.............................. 88 8.6 Infinity................................ 89 8.7 Z−: a milestone........................... 91 8.8 Selecting our natural numbers.................. 91 8.9 Appendix: closure, comprehension, and intersection . 92 Problems.................................. 93 9 Ordinals 94 9.1 The general idea of an ordinal .................. 94 9.2 Well orderings ........................... 95 9.3 Order-isomorphisms........................ 96 9.4 Von Neumann’s construction of the ordinals.......... 98 9.5 Basic properties of the ordinals.................. 98 9.6 Replacement ............................ 101 9.7 ZF−: a milestone.......................... 102 9.8 Ordinals as order-types...................... 102 9.9 Successor and limit ordinals ................... 103 Problems.................................. 104 10 Stages and ranks 105 10.1 Defining the stages as the Vas................... 105 10.2 The Transfinite Recursion Theorem(s) . 105 10.3 Basic properties of stages..................... 107 10.4 Foundation............................. 108 10.5 Z and ZF: a milestone....................... 110 10.6 Rank................................. 110 11 Replacement 112 11.1 The strength of Replacement................... 112 11.2 Extrinsic considerations about Repalement . 113 11.3 Limitation-of-size ......................... 114 11.4 Replacement and “absolute infinity” . 115 11.5 Replacement and Reflection ................... 117 Problems.................................. 120 12 Ordinal arithmetic 121 12.1 Ordinal addition.......................... 121 12.2 Using ordinal addition ...................... 124 12.3 Ordinal multiplication....................... 125 12.4 Ordinal exponentiation...................... 126 Problems.................................. 127 13 Cardinals 128 13.1 Cantor’s Principle ......................... 128 13.2 Cardinals as ordinals ....................... 128 CONTENTS 5 13.3 ZFC: a milestone.......................... 130 13.4 Finite, countable, uncountable.................. 130 13.5 Appendix: Hume’s Principle................... 133 14 Cardinal arithmetic 135 14.1 Defining the basic operations................... 135 14.2 Simplifying addition and multiplication . 137 14.3 Some simplification with cardinal exponentiation . 138 14.4 The Continuum Hypothesis ................... 139 14.5 Appendix: @-fixed-points..................... 141 Problems.................................. 143 15 Choice 144 15.1 The Tarski–Scott trick ....................... 144 15.2 Comparability and Hartogs’ Lemma . 145 15.3 The Well-Ordering Problem.................... 146 15.4 Countable choice.......................... 147 15.5 Intrinsic considerations about Choice . 149 15.6 The Banach–Tarski Paradox.................... 150 15.7 Appendix: Vitali’s Paradox.................... 152 Problems.................................. 155 Bibliography 156 Chapter 0 Prelude: history and mythology To understand the philosophical significance of set theory, it will help to have some sense of why set theory arose at all. To understand that, it will help to think a little bit about the history and mythology of mathematics. So, before we get started on discussing set theory at all, we will start with a very brief “history”. But I put this in scare-quotes, because it is very brief, extremely selective, and somewhat contestable. 0.1 Infinitesimals and differentiation Newton and Leibniz discovered the calculus (independently) at the end of the 17th century. A particularly important application of the calculus was differentiation. Roughly speaking, differentiation aims to give a notion of the “rate of change”, or gradient, of a function at a point. Here is a vivid way to illustrate the idea. Consider the function f (x) = 2 x /4 + 1/2, depicted in black below: f (x) 5 4 3 2 1 x 1 2 3 4 6 7 Suppose we want to find the gradient of the function at c = 1/2. We start by drawing a triangle whose hypotenuse approximates the gradient at that point, perhaps the red triangle above. When b is the base length of our tri- angle, its height is f (1/2 + b) − f (1/2), so that the gradient of the hypotenuse is: f (1/2 + b) − f (1/2) b So the gradient of our red triangle, with base length 3, is exactly 1. The hypotenuse of a smaller triangle, the green triangle with base length 2, gives a better approximation; its gradient is 3/4.
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