DOCSLIB.ORG

MAGIC Set Theory Lecture 4

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
MAGIC Set Theory Lecture 4 MAGIC Set theory lecture 4 David Aspero´ University of East Anglia 1 November 2018 Last week we finished with: Theorem Every well–order (L, ) is order–isomorphic to a unique ordinal. Many sets can be well–ordered in different ways (so that the corresponding well–orders have different order types). Example: ! can be well–ordered by in order type !. And it 2 can be well–ordered by putting 0 on top of every n > 0 and well-ordering ! 0 according to . This well–order has order \{ } 2 type ! + 1. [Exercise: Characterize the sets that can be well–ordered in different ways.] We have seen the ordinals ! + 1, (! + 1)+1, ((! + 1)+1)+1, etc., aka !, ! + 1, ! + 2, etc. The set consisting of all natural numbers and ! + n for every n <!is a transitive set of ordinals and therefore also an ordinal. It is called ! + !. Then we can build (! + !)+1, and so on. All these ordinals are countable (i.e., they are bijective with !). Is there an infinite ordinal which is not bijective with !? A cardinal is an ordinal such that is not bijective with any ordinal ↵<. So, each natural number is a cardinal, ! is a cardinal, but no ! + n is a cardinal for n 1. Similarly, none of ! + !, ≥ (! + !)+1, etc., are cardinals. When regarded as a cardinal, ! is also denoted (so @0 N = ! = 0). @ Notation: If X is bijective with a cardinal , we say that is the cardinality of X and write X = . | | We will see that ZFC proves that every set is bijective with an ordinal. Caveat: In a context without AC one can extend the notion of cardinals to things that are not ordinals in a perfectly meaningful way. We don’t need to do that for the moment. So, for us, at least for the moment, cardinals are ordinals. Cardinals, in our sense, i.e., cardinals that are ordinals, are sometimes also called ‘alephs’. We will see that ZFC proves that every set is bijective with an ordinal. Caveat: In a context without AC one can extend the notion of cardinals to things that are not ordinals in a perfectly meaningful way. We don’t need to do that for the moment. So, for us, at least for the moment, cardinals are ordinals. Cardinals, in our sense, i.e., cardinals that are ordinals, are sometimes also called ‘alephs’. Definition: ! , also denoted , is the first uncountable cardinal 1 @1 (in other words, the first infinite cardinal not bijective with !). Does !1 exist? Proposition (ZF) !1 exists. Proof. Say that X ! encodes a well–order if ✓ (n, m) ! ! : 2n+13m+1 X { 2 ⇥ 2 } is a well–order. Every infinite initial segment of a well–order encoded by a subset of ! can be encoded by a subset of !. Hence β = ↵ : ↵ = ot( ) for some encoded by some X ! is { ✓ } transitive and is a set since it is range(F), where F : (!) Ord is the function sending X to ot( ) if X P −! encodes and to 0 otherwise. Hence β is an ordinal. β is not countable: If f : ! β were a bijection, −! 2n+13m+1 : f (n) f (m) would be a subset of ! encoding a { 2 } well–order of order type β. But then β β, which is impossible 2 for ordinals. It is easy to see that the ordinal β in the above proof is in fact !1; that is, ! = ↵ : ↵ = ot( ) for some encoded by some X ! 1 { ✓ } [Exercise] Similarly, one can prove in ZF that there is a least cardinal strictly bigger than ! . It is called ! , or . In general, we 1 2 @2 define: Definition: Given an ordinal ↵, , also denoted ! , is the ↵–th @↵ ↵ infinite cardinal. Notation: The !↵ notation is usually used in contexts where the focus is the ordinal structure. The notation is instead @↵ used when the focus is the cardinality of this set (i.e., the property of being bijective to ). @↵ Definition: Given a cardinal , +, the successor of , is the least cardinal strictly bigger than . Hence, ( )+ = , ( )+ = , and in general, ( )+ = . @0 @1 @1 @2 @↵ @↵+1 Proposition (ZF) For every infinite cardinal Ord, + exists. 2 This follows immediately from: Theorem (Hartogs) Given any set X there is an ordinal ↵ for which there is no injective function f : ↵ X. −! Proof (of Hartogs’ Theorem): Let ↵ be the collection of ordinals β such that there is an injective function f : β X. ↵ is the −! collection of of ordinals of the form ot(Y , ), where Y X and ✓ is an order type of Y . Hence ↵ is a set by Replacement (being the image of the function with domain (X X) sending P ⇥ W to ot(W ) if W is a well–order of some Y X, and to ✓ ; otherwise), and it is trivial to see that it is transitive. Hence ↵ is an ordinal. Finally, there is no injective f : ↵ X; otherwise −! ↵ ↵, which is impossible for ordinals. 2 ⇤ Proposition (ZF) For every infinite cardinal Ord, + exists. 2 This follows immediately from: Theorem (Hartogs) Given any set X there is an ordinal ↵ for which there is no injective function f : ↵ X. −! Proof (of Hartogs’ Theorem): Let ↵ be the collection of ordinals β such that there is an injective function f : β X. ↵ is the −! collection of of ordinals of the form ot(Y , ), where Y X and ✓ is an order type of Y . Hence ↵ is a set by Replacement (being the image of the function with domain (X X) sending P ⇥ W to ot(W ) if W is a well–order of some Y X, and to ✓ ; otherwise), and it is trivial to see that it is transitive. Hence ↵ is an ordinal. Finally, there is no injective f : ↵ X; otherwise −! ↵ ↵, which is impossible for ordinals. 2 ⇤ It is easy to see that if X = is a cardinal, then the ordinal ↵ in the above proof is in fact +; that is, + = β : β = ot(R) for some well–order R { ✓ ⇥ } [Exercise] [Exercise: We have seen that exists, and so does and, in @0 @1 general, for every n. Prove that : n <! is the least @n {@n } cardinal bigger than for all n <!. Hence @n S = : n <! . In general, prove that if X is a set of @! {@n } cardinals, then X is a cardinal and is the least cardinal λ S such that λ for every X.] ≥ S 2 Theorem (Cantor–Bernstein–Schroder¨ Theorem) (ZF) For all sets X and Y , the following are equivalent. (1) X Y and Y X . | || | | || | (2) X = Y | | | | We’ll see that if f : X Y and g : Y X ire injective −! −! functions, then there is a bijection h : X Y than can be −! effectively constructed from f and g. Example: f : 2n : n ! ! and g : ! 2n : n ! { 2 } −! −!{ 2 } are injective non–surjective functions, where f is the identity and g(n)=4n for all n. We’ll see that there is a bijection h : ! 2n : n ! that can be effectively constructed from −!{ 2 } f and g. Proof: We only need to prove that (1) implies (2). For this, let f : X Y and g : Y X be injective functions. By replacing −! −! if necessary X and Y by, for example, X 0 and Y 1 , ⇥{ } ⇥{ } respectively, we may assume that X and Y are disjoint in the first place. Given c X Y , let σ be the –maximal sequence with 2 [ c ✓ domain included in Z such that σc(0)=c, σc(z + 1)=f (σc(z)) or σ (z + 1)=g(σ (z)) depending on whether σ (z) X or c c c 2 σ (z) Y , such that σ (z 1)=c¯ if c¯ X is such that c 2 c − 2 f (c¯)=σ (z) (if σ (z) Y and if there is such a c¯), and such c c 2 that σ (z 1)=c¯ if c¯ Y is such that g(c¯)=σ (z) (if c − 2 c σ (z) X and if there is such a c¯). c 2 We call σc the orbit of c. We say that σc starts in X if there is some z Z in the domain 2 of σ such that σ (z) X and such that there is no c¯ Y with c c 2 2 g(c¯)=σc(z) (so σc(z) is the first member of σc). Similarly we define ‘σc starts in Y ’. And we say that σc does not start in the remaining case (i.e., if and only if dom(σc)=Z). We also say that a set σ is an orbit if σ is (the range of) the orbit of some c X Y in the above sense. 2 [ Remark 1: Every two distinct orbits are disjoint and the orbits partition X Y . [ Remark 2: If σ is an orbit, then f σ is a bijection between σ X and σ Y if σ starts in X, • \ \ g σ is a bijection between σ Y and σ X if σ starts in • \ \ Y , and f σ is a bijection between σ X and σ Y and g σ is a • \ \ bijection between σ Y and σ X if σ does not start. \ \ Using these two remarks we can now define a bijection h : X Y −! by ‘gluing together’ suitable restrictions of f and/or of the inverse of g: Given a X: 2 If the unique orbit to which a belongs starts in X or does • not start, then h(a)=f (a). If this orbit starts in Y , let h(a) be the unique b Y such • 2 that g(b)=a. ⇤ Dual Cantor–Bernstein: For all sets X, Y , the following are equivalent: (1) X = Y | | | | (2) There is a surjection f : X Y and there is a surjection −! g : Y X.
Load more

Recommended publications
  • Calibrating Determinacy Strength in Levels of the Borel Hierarchy
    CALIBRATING DETERMINACY STRENGTH IN LEVELS OF THE BOREL HIERARCHY SHERWOOD J. HACHTMAN Abstract. We analyze the set-theoretic strength of determinacy for levels of the Borel 0 hierarchy of the form Σ1+α+3, for α < !1. Well-known results of H. Friedman and D.A. Martin have shown this determinacy to require α+1 iterations of the Power Set Axiom, but we ask what additional ambient set theory is strictly necessary. To this end, we isolate a family of Π1-reflection principles, Π1-RAPα, whose consistency strength corresponds 0 CK exactly to that of Σ1+α+3-Determinacy, for α < !1 . This yields a characterization of the levels of L by or at which winning strategies in these games must be constructed. When α = 0, we have the following concise result: the least θ so that all winning strategies 0 in Σ4 games belong to Lθ+1 is the least so that Lθ j= \P(!) exists + all wellfounded trees are ranked". x1. Introduction. Given a set A ⊆ !! of sequences of natural numbers, consider a game, G(A), where two players, I and II, take turns picking elements of a sequence hx0; x1; x2;::: i of naturals. Player I wins the game if the sequence obtained belongs to A; otherwise, II wins. For a collection Γ of subsets of !!, Γ determinacy, which we abbreviate Γ-DET, is the statement that for every A 2 Γ, one of the players has a winning strategy in G(A). It is a much-studied phenomenon that Γ -DET has mathematical strength: the bigger the pointclass Γ, the stronger the theory required to prove Γ -DET.
    [Show full text]
  • Equivalents to the Axiom of Choice and Their Uses A
    EQUIVALENTS TO THE AXIOM OF CHOICE AND THEIR USES A Thesis Presented to The Faculty of the Department of Mathematics California State University, Los Angeles In Partial Fulfillment of the Requirements for the Degree Master of Science in Mathematics By James Szufu Yang c 2015 James Szufu Yang ALL RIGHTS RESERVED ii The thesis of James Szufu Yang is approved. Mike Krebs, Ph.D. Kristin Webster, Ph.D. Michael Hoffman, Ph.D., Committee Chair Grant Fraser, Ph.D., Department Chair California State University, Los Angeles June 2015 iii ABSTRACT Equivalents to the Axiom of Choice and Their Uses By James Szufu Yang In set theory, the Axiom of Choice (AC) was formulated in 1904 by Ernst Zermelo. It is an addition to the older Zermelo-Fraenkel (ZF) set theory. We call it Zermelo-Fraenkel set theory with the Axiom of Choice and abbreviate it as ZFC. This paper starts with an introduction to the foundations of ZFC set the- ory, which includes the Zermelo-Fraenkel axioms, partially ordered sets (posets), the Cartesian product, the Axiom of Choice, and their related proofs. It then intro- duces several equivalent forms of the Axiom of Choice and proves that they are all equivalent. In the end, equivalents to the Axiom of Choice are used to prove a few fundamental theorems in set theory, linear analysis, and abstract algebra. This paper is concluded by a brief review of the work in it, followed by a few points of interest for further study in mathematics and/or set theory. iv ACKNOWLEDGMENTS Between the two department requirements to complete a master's degree in mathematics − the comprehensive exams and a thesis, I really wanted to experience doing a research and writing a serious academic paper.
    [Show full text]
  • MAGIC Set Theory Lecture 6
    MAGIC Set theory lecture 6 David Aspero´ University of East Anglia 15 November 2018 Recall: We defined (V : ↵ Ord) by recursion on Ord: ↵ 2 V = • 0 ; V = (V ) • ↵+1 P ↵ V = V : β<δ if δ is a limit ordinal. • δ { β } S (V : ↵ Ord) is called the cumulative hierarchy. ↵ 2 We saw: Proposition V↵ is transitive for every ordinal ↵. Proposition For all ↵<β, V V . ↵ ✓ β Definition For every x V , let rank(x) be the first ↵ such that 2 ↵ Ord ↵ x V . 2 2 ↵+1 S Definition For every set x, the transitive closure of x, denoted by TC(x), is X : n <! where { n } X = x S • 0 X = X • n+1 n So TC(x)=Sx x x x ... [ [ [ [ S SS SSS [Exercise: TC(x) is the –least transitive set y such that x y. ✓ ✓ In other words, TC(x)= y : y transitive, x y .] { ✓ } T Definition V denotes the class of all sets; that is, V = x : x = x . { } Definition WF = V : ↵ Ord : The class of all x such that x V for { ↵ 2 } 2 ↵ some ordinal ↵. S Note: WF is a transitive class: y x V implies y V since 2 2 ↵ 2 ↵ V↵ is transitive. Proposition If x WF, then there is some ↵ Ord such that x V . ✓ 2 ✓ ↵ Proof. Define the function F(y)=min γ y V . By the assumption { | 2 γ} on x, F is a well-defined function there. By Replacement γ y x : γ = F(y) is a set, and by Union it has a { |9 2 } supremum ↵.
    [Show full text]
  • Omega-Models of Finite Set Theory
    ω-MODELS OF FINITE SET THEORY ALI ENAYAT, JAMES H. SCHMERL, AND ALBERT VISSER Abstract. Finite set theory, here denoted ZFfin, is the theory ob- tained by replacing the axiom of infinity by its negation in the usual axiomatization of ZF (Zermelo-Fraenkel set theory). An ω-model of ZFfin is a model in which every set has at most finitely many elements (as viewed externally). Mancini and Zambella (2001) em- ployed the Bernays-Rieger method of permutations to construct a recursive ω-model of ZFfin that is nonstandard (i.e., not isomor- phic to the hereditarily finite sets Vω). In this paper we initiate the metamathematical investigation of ω-models of ZFfin. In par- ticular, we present a new method for building ω-models of ZFfin that leads to a perspicuous construction of recursive nonstandard ω-models of ZFfin without the use of permutations. Furthermore, we show that every recursive model of ZFfin is an ω-model. The central theorem of the paper is the following: Theorem A. For every graph (A, F ), where F is a set of un- ordered pairs of A, there is an ω-model M of ZFfin whose universe contains A and which satisfies the following conditions: (1) (A, F ) is definable in M; (2) Every element of M is definable in (M, a)a∈A; (3) If (A, F ) is pointwise definable, then so is M; (4) Aut(M) =∼ Aut(A, F ). Theorem A enables us to build a variety of ω-models with special features, in particular: Corollary 1. Every group can be realized as the automorphism group of an ω-model of ZFfin.
    [Show full text]
  • The Foundation Axiom and Elementary Self-Embeddings of the Universe
    THE FOUNDATION AXIOM AND ELEMENTARY SELF-EMBEDDINGS OF THE UNIVERSE ALI SADEGH DAGHIGHI, MOHAMMAD GOLSHANI, JOEL DAVID HAMKINS, AND EMIL JERˇABEK´ Abstract. We consider the role of the foundation axiom and var- ious anti-foundation axioms in connection with the nature and ex- istence of elementary self-embeddings of the set-theoretic universe. We shall investigate the role of the foundation axiom and the vari- ous anti-foundation axioms in connection with the nature and exis- tence of elementary self-embeddings of the set-theoretic universe. All the standard proofs of the well-known Kunen inconsistency [Kun78], for example, the theorem asserting that there is no nontrivial elemen- tary embedding of the set-theoretic universe to itself, make use of the axiom of foundation (see [Kan04, HKP12]), and this use is essential, assuming that ZFC is consistent, because there are models of ZFC−f that admit nontrivial elementary self-embeddings and even nontrivial definable automorphisms. Meanwhile, a fragment of the Kunen incon- sistency survives without foundation as the claim in ZFC−f that there is no nontrivial elementary self-embedding of the class of well-founded sets. Nevertheless, some of the commonly considered anti-foundational theories, such as the Boffa theory BAFA, prove outright the existence of nontrivial automorphisms of the set-theoretic universe, thereby refuting the Kunen assertion in these theories. On the other hand, several other common anti-foundational theories, such as Aczel’s anti-foundational −f −f arXiv:1311.0814v2 [math.LO] 13 Feb 2014 theory ZFC + AFA and Scott’s theory ZFC + SAFA, reach the op- posite conclusion by proving that there are no nontrivial elementary The second author’s research has been supported by a grant from IPM (No.
    [Show full text]
  • A QUICK INTRODUCTION to BASIC SET THEORY This Note Is Intended
    A QUICK INTRODUCTION TO BASIC SET THEORY ANUSH TSERUNYAN This note is intended to quickly introduce basic notions in set theory to (undergraduate) students in mathematics not necessarily specializing in logic, thus requiring no background. It uses the sections on well-orderings, ordinals and cardinal arithmetic of [Kun83], as well as some parts from [Mos06]. Contents 1. Zermelo–Fraenkel set theory 2 2. Well-orderings 6 3. Ordinals 7 4. Transfinite induction 9 5. The set of natural numbers 10 6. Equinumerosity 11 7. Finite/infinite sets and Axiom of Choice 14 8. Cardinals and cardinality 16 9. Uncountable cardinals and the Continuum Hypothesis 16 10. Cofinality and Konig˝ ’s theorem 17 11. Classes 18 Exercises 19 References 21 1 1. Zermelo–Fraenkel set theory The language of ZFC. ZF stands for Zermelo–Fraenkel set theory and ZFC stands for Zermelo– Fraenkel set theory with Choice (the latter being an extra axiom added to ZF). To describe the axioms of ZFC we need to fix a language (formally speaking, a first order logic language). The language of ZF (same for ZFC) has only one special symbol, namely the binary relation symbol 2, in addition to the standard symbols that every other first order language has: the symbols =; :; ^; _; !; 8; 9; (; ) and variables x0; x1; x2; :::. To keep things informal and easy to read, we also use other letters such as x; y; z or even A, B, C, F for variables, as well as extra symbols like ,; <; $, etc. Using these symbols and variables we form statements about sets such as 8x(x = x), 9x(x , x), 9x(x 2 y ^ y < z).
    [Show full text]
  • Sharp Sets of Permutations
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 111, 22C-247 (1987) Sharp Sets of Permutations P. J. CAMERON* Merton College, Oxford OX1 4JD, United Kingdom AND M. DEZA AND P. FRANKL CNRS, Paris, France Communicated by Peter hi. Neumann Received January 29, 1984 A set of permutations is sharp if its cardinality is the product of the distinct non-zero Hamming distances between pairs of permutations in the set. We give a number of new results and constructions for sharp sets and groups and for the more special geometric sets and groups, using a mixture of algebraic and combinatorial techniques. 0 1987 Academx Press, Inc. 1. IN~~DUCTI~N Sharp sets of permutations arise in the study of extremal problems for sets of permutations with prescribed Hamming distances. In this paper, we give some new constructions and results about sharp sets and groups and the closely related permutation geometries. A more precise description of the results will be given after the relevant concepts have been defined. Let S, be the group of permutations of { 1, .. n}; for g E S,, let tix( g) denote the number of fixed points of g. Given any non-empty subset L = { 10,.*.> l,- I} of (0, .. n-l}, (where we assume that I,< ... <IS--), a sub- set A of S, is called L-intersecting if, for all g, h EA with g # h, we have fix(g-‘h) E L. (Note that n-fix(g-‘h) is the Hamming distance between g and h.) Denote by m(n, L) the maximum cardinality of an L-intersecting subset of S,, and by mg(n, L) that of an L-intersecting subgroup.
    [Show full text]
  • Constructive Zermelo-Fraenkel Set Theory, Power Set, and the Calculus of Constructions
    This is a repository copy of Constructive zermelo-fraenkel set theory, power set, and the calculus of constructions. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/75182/ Book Section: Rathjen, M (2012) Constructive zermelo-fraenkel set theory, power set, and the calculus of constructions. In: Dybjer, P, Lindström, S, Palmgren, E and Sundholm, G, (eds.) Epistemology versus ontology: Essays on the philosophy and foundations of mathematics in honour of Per Martin-Löf. Logic, Epistemology, and the Unity of Science, 27 . Springer , Dordrecht, Netherlands , 313 - 349. ISBN 9789400744356 https://doi.org/10.1007/978-94-007-4435-6 Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ Constructive Zermelo-Fraenkel Set Theory, Power Set, and the Calculus of Constructions Michael Rathjen∗ Department of Pure Mathematics, University of Leeds Leeds LS2 9JT, United Kingdom [email protected] May 4, 2012 Abstract Full intuitionistic Zermelo-Fraenkel set theory, IZF, is obtained from constructive Zermelo- Fraenkel set theory, CZF, by adding the full separation axiom scheme and the power set axiom.
    [Show full text]
  • ON the NON–EXISTENCE of SHARPLY TRANSITIVE SETS of PERMUTATIONS in CERTAIN FINITE PERMUTATION GROUPS Peter Müller Gábor P. N
    Advances in Mathematics of Communications doi:10.3934/amc.2011.5.303 Volume 5, No. 2, 2011, 303{308 ON THE NON{EXISTENCE OF SHARPLY TRANSITIVE SETS OF PERMUTATIONS IN CERTAIN FINITE PERMUTATION GROUPS Peter Muller¨ Institut f¨urMathematik, Universit¨atW¨urzburg Campus Hubland Nord, 97074 W¨urzburg,Germany Gabor´ P. Nagy Bolyai Institute, University of Szeged Aradi v´ertan´uktere 1, H-6720 Szeged, Hungary (Communicated by Reinhard Laue) Abstract. In this short note we present a simple combinatorial trick which can be effectively applied to show the non{existence of sharply transitive sets of permutations in certain finite permutation groups. 1. Introduction A permutation code (or array) of length n and distance d is a set S of per- mutations of some fixed set Ω of n symbols such that the Hamming distance between each distinct x; y 2 S is at least d, see [4]. By elementary counting, one has jSj ≤ n(n − 1) ··· d and equality holds if and only if for any two tuples (x1; : : : ; xn−d+1), (y1; : : : ; yn−d+1) of distinct symbols, there is a unique element s s s 2 S with x1 = y1; : : : ; xn−d+1 = yn−d+1. Such sets of permutations are called sharply t{transitive, where t = n − d + 1. It is well known that sharply 1{ and 2{transitive sets of permutations correspond to Latin squares and affine planes, respectively [3]. In general, there are very few results on permutation codes and there is a large gap between the lower and upper estimates for jSj; see [10], [9].
    [Show full text]
  • An Introduction to Set Theory
    AN INTRODUCTION TO SET THEORY Professor William A. R. Weiss November 21, 2014 2 Contents 0 Introduction 7 1 LOST 11 2 FOUND 23 3 The Axioms of Set Theory 29 4 The Natural Numbers 37 5 The Ordinal Numbers 47 6 Relations and Orderings 59 7 Cardinality 69 8 What's So Real About The Real Numbers? 79 9 Ultrafilters Are Useful 87 3 4 CONTENTS 10 The Universe 97 11 Reflection 103 12 Elementary Submodels 123 13 Constructibility 139 14 Appendices 155 .1 The Axioms of ZFC . 155 .2 Tentative Axioms . 156 CONTENTS 5 Preface These notes for a graduate course in set theory are on their way to be- coming a book. They originated as handwritten notes in a course at the University of Toronto given by Prof. William Weiss. Cynthia Church pro- duced the first electronic copy in December 2002. James Talmage Adams produced a major revision in February 2005. The manuscript has seen many changes since then, often due to generous comments by students, each of whom I here thank. Chapters 1 to 11 are now close to final form. Chapters 12 and 13 are quite readable, but should not be considered as a final draft. One more chapter will be added. 6 CONTENTS Chapter 0 Introduction Set Theory is the true study of infinity. This alone assures the subject of a place prominent in human culture. But even more, Set Theory is the milieu in which mathematics takes place today. As such, it is expected to provide a firm foundation for all the rest of mathematics.
    [Show full text]
  • Sharply (N − 2)-Transitive Sets of Permutations
    Iranian Journal of Mathematical Sciences and Informatics Vol. 15, No. 2 (2020), pp 183-190 DOI: 10.29252/ijmsi.15.2.183 Sharply (n − 2)-transitive Sets of Permutations Moharram N. Iradmusa Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran. E-mail: [email protected] Abstract. Let Sn be the symmetric group on the set [n] = f1; 2; : : : ; ng. For g 2 Sn let fix(g) denote the number of fixed points of g. A subset S of Sn is called t-transitive if for any two t-tuples (x1; x2; : : : ; xt) and (y1; y2; : : : ; yt) of distinct elements of [n], there exists g 2 S such that g ≤ ≤ xi = yi for any 1 i t and additionally S is called sharply t-transitive if for any given pair of t-tuples, exactly one element of S carries the first to the second. In addition, a subset S of Sn is called t-intersecting if fix(h−1g) ≥ t for any two distinct permutations h and g of S. In this paper, we prove that there are only two sharply (n − 2)-transitive subsets of Sn and finally we establish some relations between sharply k-transitive subsets and t-intersecting subsets of Sn where k; t 2 Z and 0 ≤ t ≤ k ≤ n. Keywords: Symmetric group, Sharply transitive set of permutations, Cay- ley graph. 2010 Mathematics Subject Classification: 20B20, 20B30. 1. Introduction Let Sn be the symmetric group on the set [n] = f 1; 2; : : : ; ng . For each g 2 Sn, a point x is called a fixed point of g if g(x) = x and fix(g) denotes the number of fixed points of g.
    [Show full text]
  • Undergraduate Logic Sequence: the Notes
    Undergraduate logic sequence: the notes November 4, 2015 ii Contents 1 Zermelo{Fraenkel set theory 1 1.1 Historical context . 1 1.2 The language of the theory . 3 1.3 The most basic axioms . 4 1.4 Axiom schema of Comprehension . 6 1.5 Relations and functions . 7 1.6 Axiom of Infinity . 9 1.7 Axiom of Choice . 10 1.8 Axiom schema of Replacement . 11 1.9 Axiom of Regularity . 12 2 Numbers, Ordinals, and Cardinals 15 2.1 Von Neumann's natural numbers . 15 2.2 Finite and infinite sets . 18 2.3 Inductive and Recursive Definability . 21 2.4 Models of Set Theory . 24 2.5 The Hereditarily Finite Sets . 26 2.6 Cardinality . 28 2.7 Countable and uncountable sets . 31 3 The transfinite 35 3.1 Ordinals . 35 3.2 Transfinite induction and recursion . 38 3.3 Ordinal Arithmetic . 44 3.4 Ordinals and Well-Orderings . 49 3.5 Equivalent versions of the Axiom of Choice . 51 3.6 Applications of choice . 53 3.7 Cardinal numbers . 55 3.8 Models of Set Theory, Revisited . 57 4 Descriptive set theory 59 4.1 Rational and real numbers . 59 4.2 Topological spaces . 61 4.3 Polish spaces . 64 iii iv CONTENTS 4.4 Universality theorems . 67 4.5 Borel sets . 69 4.6 Analytic sets . 72 4.7 Lebesgue's mistake . 74 4.8 Suslin's fix . 76 5 First order logic 79 5.1 Propositional logic . 79 5.2 Syntax . 84 5.3 Semantics . 87 5.4 Completeness theorem . 88 Chapter 1 Zermelo{Fraenkel set theory 1.1 Historical context In 19th century, mathematicians produced a great number of sophisticated the- orems and proofs.
    [Show full text]