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# SetTheory - Foundations of Mathematics I Set Theory(only a...

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Foundations of Mathematics I Set Theory (only a draft) Ali Nesin Mathematics Department Istanbul Bilgi University Ku¸ stepe S ¸i¸ sli Istanbul Turkey [email protected] February 12, 2004

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Contents I Naive Set Theory 7 1 Basic Concepts and Examples 11 1.1 Sets, Subsets and Emptyset . . . . . . . . . . . . . . . . . . . . . 11 1.2 Notes on Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3 Number Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4 Subsets Defined By a Property . . . . . . . . . . . . . . . . . . . 20 1.5 Sets of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.6 Parametrized Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2 Operations with Sets 25 2.1 Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3 Union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4 Cartesian Product of Two Sets . . . . . . . . . . . . . . . . . . . 29 3 Functions 31 3.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 More On Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 Binary Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4 Operations with Functions . . . . . . . . . . . . . . . . . . . . . . 37 3.5 Injections, Surjections, Bijections . . . . . . . . . . . . . . . . . . 38 3.5.1 Injections . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.5.2 Surjections . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.5.3 Bijections . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.6 Sym( X ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.7 Families, Sequences and Cartesian Products . . . . . . . . . . . . 44 4 Relations 47 4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.2 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . 48 4.3 Partial Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.4 Total Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5 Induction 57 3

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4 CONTENTS 6 Bijections, revisited 61 6.1 Schr¨ oder-Bernstein’s Theorem . . . . . . . . . . . . . . . . . . . . 61 6.2 Examples of Sets in Bijection . . . . . . . . . . . . . . . . . . . . 61 7 Some Automorphism Groups 63 7.1 Binary Unirelational Structures . . . . . . . . . . . . . . . . . . . 63 7.2 Automorphism Groups of Graphs . . . . . . . . . . . . . . . . . . 64 7.3 Geometric Automorphism Groups . . . . . . . . . . . . . . . . . 65 7.4 Back and Forth Argument . . . . . . . . . . . . . . . . . . . . . . 67 7.4.1 Dense Total Orderings . . . . . . . . . . . . . . . . . . . . 67 7.4.2 Random Graphs . . . . . . . . . . . . . . . . . . . . . . . 68 8 Formulae 69 9 Miscellaneous Exercises 71 II Axiomatic Set Theory 73 10 Basics 75 10.1 Russell’s Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 10.2 Easy Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 10.3 Slightly More Complicated Axioms . . . . . . . . . . . . . . . . . 81 10.4 Cartesian Product of Two Sets . . . . . . . . . . . . . . . . . . . 82 10.5 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 11 Natural Numbers 87 11.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . 87 11.2 Well-ordering on ω . . . . . . . . . . . . . . . . . . . . . . . . . . 88 11.3 Peano’s Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 11.4 Addition of Natural Numbers . . . . . . . . . . . . . . . . . . . . 90 11.5 Multiplication of Natural Numbers . . . . . . . . . . . . . . . . . 92 11.6 Well-Ordering of Natural Numbers . . . . . . . . . . . . . . . . . 94 11.7 Finite and Infinite Sets . . . . . . . . . . . . . . . . . . . . . . . . 96 11.8 Functions Defined by Induction . . . . . . . . . . . . . . . . . . . 96 11.9 Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 11.10Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 11.11Uniqueness of N . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 12 Integers 101 12.1 Definition of The Set Z of Integers . . . . . . . . . . . . . . . . . 101 12.2 Operations +, - and × on Z . . . . . . . . . . . . . . . . . . . . 102 12.3 Ordering on Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 12.4 Embedding of ( N , + , × , < ) in ( Z , + , × , < ) . . . . . . . . . . . . . 106 12.5 Additive Structure of Z . . . . . . . . . . . . . . . . . . . . . . . 107 12.6 Multiplicative Structure of Z . . . . . . . . . . . . . . . . . . . . 108 12.7 Ordering on Z , Revisited . . . . . . . . . . . . . . . . . . . . . . . 108
CONTENTS 5 12.8 Divisibility and Subgroups . . . . . . . . . . . . . . . . . . . . . . 109 12.9 Euclidean Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 112 12.10Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 12.11Chinese Remainder Theorem . . . . . . . . . . . . . . . . . . . . 113 12.12Subgroups of Z Z . . . . . . . . . . . . . . . . . . . . . . . . . . 113 13 Rational Numbers 115 13.1 Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 115 13.2 Some Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . 118 14 Real Numbers 119 15 Well-Ordered Sets 121 15.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . 121 15.2 Transfinite Induction . . . . . . . . . . . . . . . . . . . . . . . . . 122 15.3 Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 16 Ordinals 125 16.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 16.2 Axiom of Replacement . . . . . . . . . . . . . . . . . . . . . . . . 126 16.3 Classification of Well-Ordered Sets . . . . . . . . . . . . . . . . . 126 16.4 Addition of Ordinals . . . . . . . . . . . . . . . . . . . . . . . . . 127 16.5 Multiplication of Ordinals . . . . . . . . . . . . . . . . . . . . . . 127 17 Cardinals 129 17.1 Addition of Cardinals . . . . . . . . . . . . . . . . . . . . . . . . 129 17.2 Multiplication of Cardinals . . . . . . . . . . . . . . . . . . . . . 129 17.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 17.4 Final Exam of Math 112, May 2003 . . . . . . . . . . . . . . . . 129 18 Axiom of Choice and Zorn’s Lemma 131 18.1 Zorn’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 18.2 Some Consequences of Zorn’s Lemma . . . . . . . . . . . . . . . 132 18.2.1 Finite and Infinite Sets . . . . . . . . . . . . . . . . . . . 132 18.2.2 K¨onig’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . 132 18.2.3 Some Unexpected Consequences of Zorn’s Lemma . . . . 132 19 Axioms of Set Theory – ZFC 133 20 V = L 135 21 Continuum Hypothesis 137 22 Banach-Tarski Paradox 139 23 First Order Structures 141

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6 CONTENTS 24 Ultraproducts and Ultrafilters 143 24.1 Nonstandard Models of N . . . . . . . . . . . . . . . . . . . . . . 143 24.2 Nonstandard Models of R . . . . . . . . . . . . . . . . . . . . . . 143 25 Dimension Theory 145 26 Exams 147 26.1 First Semester Midterm, November 2002 . . . . . . . . . . . . . . 147 26.2 First Semester Final, January 2003
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