DOCSLIB.ORG

Determinacy of Infinite Games

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Determinacy of Infinite Games Introduction Open games Determinacy and the Axiom of Choice Axiom of Determinacy The Perfect Subset Property Determinacy of Innite Games Regula Krapf University of Bonn 3. November 2014 Regula Krapf Determinacy of Innite Games Introduction Open games Determinacy and the Axiom of Choice Axiom of Determinacy The Perfect Subset Property Contents 1 Introduction 2 Open games 3 Determinacy and the Axiom of Choice 4 Axiom of Determinacy 5 The Perfect Subset Property Regula Krapf Determinacy of Innite Games Introduction Open games Determinacy and the Axiom of Choice Axiom of Determinacy The Perfect Subset Property What are innite games? Player I chooses x0 2 !, Player II chooses x1 2 ! and so on. Player I: x0 x2 x4 ... Player II: x1 x3 This yields a sequence of natural numbers x = hxn j n 2 !i, i.e. a real. Alternative: all xn 2 2 = f0; 1g. Regula Krapf Determinacy of Innite Games Introduction Open games Determinacy and the Axiom of Choice Axiom of Determinacy The Perfect Subset Property What are innite games? Player I chooses x0 2 !, Player II chooses x1 2 ! and so on. Player I: x0 x2 x4 ... Player II: x1 x3 This yields a sequence of natural numbers x = hxn j n 2 !i, i.e. a real. Alternative: all xn 2 2 = f0; 1g. Regula Krapf Determinacy of Innite Games Introduction Open games Determinacy and the Axiom of Choice Axiom of Determinacy The Perfect Subset Property What are innite games? Player I chooses x0 2 !, Player II chooses x1 2 ! and so on. Player I: x0 x2 x4 ... Player II: x1 x3 This yields a sequence of natural numbers x = hxn j n 2 !i, i.e. a real. Alternative: all xn 2 2 = f0; 1g. Regula Krapf Determinacy of Innite Games Introduction Open games Determinacy and the Axiom of Choice Axiom of Determinacy The Perfect Subset Property How to win an innite game Given a payo set A ⊆ !! (resp. !2), Player I wins, if x 2 A, otherwise Player II wins. A winning strategy for Player I is a map [ 2n σ : ! ! ! n2! such that Player I wins, if he plays as follows: 1 x0 = σ(;); 2 Given x0;:::; x2n+1, Player I plays x2n+2 = σ(hx0;:::; x2n+1i). Similarly for Player II. Regula Krapf Determinacy of Innite Games Introduction Open games Determinacy and the Axiom of Choice Axiom of Determinacy The Perfect Subset Property How to win an innite game Given a payo set A ⊆ !! (resp. !2), Player I wins, if x 2 A, otherwise Player II wins. A winning strategy for Player I is a map [ 2n σ : ! ! ! n2! such that Player I wins, if he plays as follows: 1 x0 = σ(;); 2 Given x0;:::; x2n+1, Player I plays x2n+2 = σ(hx0;:::; x2n+1i). Similarly for Player II. Regula Krapf Determinacy of Innite Games Introduction Open games Determinacy and the Axiom of Choice Axiom of Determinacy The Perfect Subset Property How to win an innite game Given a payo set A ⊆ !! (resp. !2), Player I wins, if x 2 A, otherwise Player II wins. A winning strategy for Player I is a map [ 2n σ : ! ! ! n2! such that Player I wins, if he plays as follows: 1 x0 = σ(;); 2 Given x0;:::; x2n+1, Player I plays x2n+2 = σ(hx0;:::; x2n+1i). Similarly for Player II. Regula Krapf Determinacy of Innite Games Introduction Open games Determinacy and the Axiom of Choice Axiom of Determinacy The Perfect Subset Property How to win an innite game Given a payo set A ⊆ !! (resp. !2), Player I wins, if x 2 A, otherwise Player II wins. A winning strategy for Player I is a map [ 2n σ : ! ! ! n2! such that Player I wins, if he plays as follows: 1 x0 = σ(;); 2 Given x0;:::; x2n+1, Player I plays x2n+2 = σ(hx0;:::; x2n+1i). Similarly for Player II. Regula Krapf Determinacy of Innite Games Introduction Open games Determinacy and the Axiom of Choice Axiom of Determinacy The Perfect Subset Property How to win an innite game Given a payo set A ⊆ !! (resp. !2), Player I wins, if x 2 A, otherwise Player II wins. A winning strategy for Player I is a map [ 2n σ : ! ! ! n2! such that Player I wins, if he plays as follows: 1 x0 = σ(;); 2 Given x0;:::; x2n+1, Player I plays x2n+2 = σ(hx0;:::; x2n+1i). Similarly for Player II. Regula Krapf Determinacy of Innite Games Introduction Open games Determinacy and the Axiom of Choice Axiom of Determinacy The Perfect Subset Property So let's play... Examples: Is there a winning strategy for the games where A ⊆ !! is given by... all eventually periodic sequences the set of all sequences where every natural number occurs at least once the set of all sequences where every natural number occurs innitely often a countable set? Regula Krapf Determinacy of Innite Games Introduction Open games Determinacy and the Axiom of Choice Axiom of Determinacy The Perfect Subset Property So let's play... Examples: Is there a winning strategy for the games where A ⊆ !! is given by... all eventually periodic sequences the set of all sequences where every natural number occurs at least once the set of all sequences where every natural number occurs innitely often a countable set? Regula Krapf Determinacy of Innite Games Introduction Open games Determinacy and the Axiom of Choice Axiom of Determinacy The Perfect Subset Property So let's play... Examples: Is there a winning strategy for the games where A ⊆ !! is given by... all eventually periodic sequences the set of all sequences where every natural number occurs at least once the set of all sequences where every natural number occurs innitely often a countable set? Regula Krapf Determinacy of Innite Games Introduction Open games Determinacy and the Axiom of Choice Axiom of Determinacy The Perfect Subset Property So let's play... Examples: Is there a winning strategy for the games where A ⊆ !! is given by... all eventually periodic sequences the set of all sequences where every natural number occurs at least once the set of all sequences where every natural number occurs innitely often a countable set? Regula Krapf Determinacy of Innite Games Introduction Open games Determinacy and the Axiom of Choice Axiom of Determinacy The Perfect Subset Property What is a winning strategy? Denition A position according to a strategy σ is a partial play such that the last play was played by the player using σ. Lemma Let A be a payo set. Assume that Player I (II) has a winning strategy σ. If x 2= A (x 2 A), then there is a position p ⊆ x according to σ such that whatever Player II (I) plays next, the consequent position according to σ is not contained in x. Regula Krapf Determinacy of Innite Games Introduction Open games Determinacy and the Axiom of Choice Axiom of Determinacy The Perfect Subset Property What is a winning strategy? Denition A position according to a strategy σ is a partial play such that the last play was played by the player using σ. Lemma Let A be a payo set. Assume that Player I (II) has a winning strategy σ. If x 2= A (x 2 A), then there is a position p ⊆ x according to σ such that whatever Player II (I) plays next, the consequent position according to σ is not contained in x. Regula Krapf Determinacy of Innite Games Introduction Open games Determinacy and the Axiom of Choice Axiom of Determinacy The Perfect Subset Property Baire space Let's dene a topology on !!, the so-called Baire space. Basic clopen sets are given by ! Is = fx 2 ! j s ⊆ xg for s 2 <!!. i.e. all sequences that start with the nite sequence s. Regula Krapf Determinacy of Innite Games Introduction Open games Determinacy and the Axiom of Choice Axiom of Determinacy The Perfect Subset Property Baire space Let's dene a topology on !!, the so-called Baire space. Basic clopen sets are given by ! Is = fx 2 ! j s ⊆ xg for s 2 <!!. i.e. all sequences that start with the nite sequence s. Regula Krapf Determinacy of Innite Games Introduction Open games Determinacy and the Axiom of Choice Axiom of Determinacy The Perfect Subset Property Are all open sets determined? Theorem (Gale-Stewart) All open sets are determined. Proof (Sketch). Let A ⊆ !! be open. Assume Player I has no winning strategy. Then at the position hx0i there must be x1 such that Player I has no winning strategy at position hx0; x1i etc. In general, Player II has a strategy τ such that at every position hx0; ··· ; x2n−1i Player I has no winning strategy. Then τ is a winning strategy for Player II: Otherwise let x 2 A be a play according to τ. Since A is open, there exists s 2 <!! such that x 2 Is ⊆ A. But then Player I has a winning strategy at all positions later than s. Contradiction. a This uses the axiom of choice! Regula Krapf Determinacy of Innite Games Introduction Open games Determinacy and the Axiom of Choice Axiom of Determinacy The Perfect Subset Property Are all open sets determined? Theorem (Gale-Stewart) All open sets are determined. Proof (Sketch). Let A ⊆ !! be open. Assume Player I has no winning strategy. Then at the position hx0i there must be x1 such that Player I has no winning strategy at position hx0; x1i etc. In general, Player II has a strategy τ such that at every position hx0; ··· ; x2n−1i Player I has no winning strategy. Then τ is a winning strategy for Player II: Otherwise let x 2 A be a play according to τ.
Load more

Recommended publications
  • GTI Diagonalization
    GTI Diagonalization A. Ada, K. Sutner Carnegie Mellon University Fall 2017 1 Comments Cardinality Infinite Cardinality Diagonalization Personal Quirk 1 3 “Theoretical Computer Science (TCS)” sounds distracting–computers are just a small part of the story. I prefer Theory of Computation (ToC) and will refer to that a lot. ToC: computability theory complexity theory proof theory type theory/set theory physical realizability Personal Quirk 2 4 To my mind, the exact relationship between physics and computation is an absolutely fascinating open problem. It is obvious that the standard laws of physics support computation (ignoring resource bounds). There even are people (Landauer 1996) who claim . this amounts to an assertion that mathematics and com- puter science are a part of physics. I think that is total nonsense, but note that Landauer was no chump: in fact, he was an excellent physicists who determined the thermodynamical cost of computation and realized that reversible computation carries no cost. At any rate . Note the caveat: “ignoring resource bounds.” Just to be clear: it is not hard to set up computations that quickly overpower the whole (observable) physical universe. Even a simple recursion like this one will do. A(0, y) = y+ A(x+, 0) = A(x, 1) A(x+, y+) = A(x, A(x+, y)) This is the famous Ackermann function, and I don’t believe its study is part of physics. And there are much worse examples. But the really hard problem is going in the opposite direction: no one knows how to axiomatize physics in its entirety, so one cannot prove that all physical processes are computable.
    [Show full text]
  • Cardinality of Accumulation Points of Infinite Sets 1 Introduction
    International Mathematical Forum, Vol. 11, 2016, no. 11, 539 - 546 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2016.6224 Cardinality of Accumulation Points of Infinite Sets A. Kalapodi CTI Diophantus, Computer Technological Institute & Press University Campus of Patras, 26504 Patras, Greece Copyright c 2016 A. Kalapodi. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract One of the fundamental theorems in real analysis is the Bolzano- Weierstrass property according to which every bounded infinite set of real numbers has an accumulation point. Since this theorem essentially asserts the completeness of the real numbers, the notion of accumulation point becomes substantial. This work provides an efficient number of examples which cover every possible case in the study of accumulation points, classifying the different sizes of the derived set A0 and of the sets A \ A0, A0 n A, for an infinite set A. Mathematics Subject Classification: 97E60, 97I30 Keywords: accumulation point; derived set; countable set; uncountable set 1 Introduction The \accumulation point" is a mathematical notion due to Cantor ([2]) and although it is fundamental in real analysis, it is also important in other areas of pure mathematics, such as the study of metric or topological spaces. Following the usual notation for a metric space (X; d), we denote by V (x0;") = fx 2 X j d(x; x0) < "g the open sphere of center x0 and radius " and by D(x0;") the set V (x0;") n fx0g.
    [Show full text]
  • Axiomatic Set Teory P.D.Welch
    Axiomatic Set Teory P.D.Welch. August 16, 2020 Contents Page 1 Axioms and Formal Systems 1 1.1 Introduction 1 1.2 Preliminaries: axioms and formal systems. 3 1.2.1 The formal language of ZF set theory; terms 4 1.2.2 The Zermelo-Fraenkel Axioms 7 1.3 Transfinite Recursion 9 1.4 Relativisation of terms and formulae 11 2 Initial segments of the Universe 17 2.1 Singular ordinals: cofinality 17 2.1.1 Cofinality 17 2.1.2 Normal Functions and closed and unbounded classes 19 2.1.3 Stationary Sets 22 2.2 Some further cardinal arithmetic 24 2.3 Transitive Models 25 2.4 The H sets 27 2.4.1 H - the hereditarily finite sets 28 2.4.2 H - the hereditarily countable sets 29 2.5 The Montague-Levy Reflection theorem 30 2.5.1 Absoluteness 30 2.5.2 Reflection Theorems 32 2.6 Inaccessible Cardinals 34 2.6.1 Inaccessible cardinals 35 2.6.2 A menagerie of other large cardinals 36 3 Formalising semantics within ZF 39 3.1 Definite terms and formulae 39 3.1.1 The non-finite axiomatisability of ZF 44 3.2 Formalising syntax 45 3.3 Formalising the satisfaction relation 46 3.4 Formalising definability: the function Def. 47 3.5 More on correctness and consistency 48 ii iii 3.5.1 Incompleteness and Consistency Arguments 50 4 The Constructible Hierarchy 53 4.1 The L -hierarchy 53 4.2 The Axiom of Choice in L 56 4.3 The Axiom of Constructibility 57 4.4 The Generalised Continuum Hypothesis in L.
    [Show full text]
  • 17 Axiom of Choice
    Math 361 Axiom of Choice 17 Axiom of Choice De¯nition 17.1. Let be a nonempty set of nonempty sets. Then a choice function for is a function f sucFh that f(S) S for all S . F 2 2 F Example 17.2. Let = (N)r . Then we can de¯ne a choice function f by F P f;g f(S) = the least element of S: Example 17.3. Let = (Z)r . Then we can de¯ne a choice function f by F P f;g f(S) = ²n where n = min z z S and, if n = 0, ² = min z= z z = n; z S . fj j j 2 g 6 f j j j j j 2 g Example 17.4. Let = (Q)r . Then we can de¯ne a choice function f as follows. F P f;g Let g : Q N be an injection. Then ! f(S) = q where g(q) = min g(r) r S . f j 2 g Example 17.5. Let = (R)r . Then it is impossible to explicitly de¯ne a choice function for . F P f;g F Axiom 17.6 (Axiom of Choice (AC)). For every set of nonempty sets, there exists a function f such that f(S) S for all S . F 2 2 F We say that f is a choice function for . F Theorem 17.7 (AC). If A; B are non-empty sets, then the following are equivalent: (a) A B ¹ (b) There exists a surjection g : B A. ! Proof. (a) (b) Suppose that A B.
    [Show full text]
  • Worksheet: Cardinality, Countable and Uncountable Sets
    Math 347 Worksheet: Cardinality A.J. Hildebrand Worksheet: Cardinality, Countable and Uncountable Sets • Key Tool: Bijections. • Definition: Let A and B be sets. A bijection from A to B is a function f : A ! B that is both injective and surjective. • Properties of bijections: ∗ Compositions: The composition of bijections is a bijection. ∗ Inverse functions: The inverse function of a bijection is a bijection. ∗ Symmetry: The \bijection" relation is symmetric: If there is a bijection f from A to B, then there is also a bijection g from B to A, given by the inverse of f. • Key Definitions. • Cardinality: Two sets A and B are said to have the same cardinality if there exists a bijection from A to B. • Finite sets: A set is called finite if it is empty or has the same cardinality as the set f1; 2; : : : ; ng for some n 2 N; it is called infinite otherwise. • Countable sets: A set A is called countable (or countably infinite) if it has the same cardinality as N, i.e., if there exists a bijection between A and N. Equivalently, a set A is countable if it can be enumerated in a sequence, i.e., if all of its elements can be listed as an infinite sequence a1; a2;::: . NOTE: The above definition of \countable" is the one given in the text, and it is reserved for infinite sets. Thus finite sets are not countable according to this definition. • Uncountable sets: A set is called uncountable if it is infinite and not countable. • Two famous results with memorable proofs.
    [Show full text]
  • The Set of All Countable Ordinals: an Inquiry Into Its Construction, Properties, and a Proof Concerning Hereditary Subcompactness
    W&M ScholarWorks Undergraduate Honors Theses Theses, Dissertations, & Master Projects 5-2009 The Set of All Countable Ordinals: An Inquiry into Its Construction, Properties, and a Proof Concerning Hereditary Subcompactness Jacob Hill College of William and Mary Follow this and additional works at: https://scholarworks.wm.edu/honorstheses Part of the Mathematics Commons Recommended Citation Hill, Jacob, "The Set of All Countable Ordinals: An Inquiry into Its Construction, Properties, and a Proof Concerning Hereditary Subcompactness" (2009). Undergraduate Honors Theses. Paper 255. https://scholarworks.wm.edu/honorstheses/255 This Honors Thesis is brought to you for free and open access by the Theses, Dissertations, & Master Projects at W&M ScholarWorks. It has been accepted for inclusion in Undergraduate Honors Theses by an authorized administrator of W&M ScholarWorks. For more information, please contact [email protected]. The Set of All Countable Ordinals: An Inquiry into Its Construction, Properties, and a Proof Concerning Hereditary Subcompactness A thesis submitted in partial fulfillment of the requirement for the degree of Bachelor of Science with Honors in Mathematics from the College of William and Mary in Virginia, by Jacob Hill Accepted for ____________________________ (Honors, High Honors, or Highest Honors) _______________________________________ Director, Professor David Lutzer _________________________________________ Professor Vladimir Bolotnikov _________________________________________ Professor George Rublein _________________________________________
    [Show full text]
  • A Search for a Constructivist Approach for Understanding the Uncountable Set P(N) Cindy Stenger University of North Alabama
    CORE Metadata, citation and similar papers at core.ac.uk Provided by DigitalCommons@Florida International University Florida International University FIU Digital Commons School of Education and Human Development College of Arts, Sciences & Education Faculty Publications 3-2008 A Search for a Constructivist Approach for Understanding the Uncountable Set P(N) Cindy Stenger University of North Alabama Kirk Weller University of Michigan-Flint Ilana Arnon Center for Educational Technology, Tel Aviv Ed Dubinsky Florida International University Draga Vidakovic Georgia State University Follow this and additional works at: http://digitalcommons.fiu.edu/education_fac Part of the Education Commons Recommended Citation Stenger, Cindy; Weller, Kirk; Arnon, Ilana; Dubinsky, Ed; and Vidakovic, Draga, "A Search for a Constructivist Approach for Understanding the Uncountable Set P(N)" (2008). School of Education and Human Development Faculty Publications. 1. http://digitalcommons.fiu.edu/education_fac/1 This work is brought to you for free and open access by the College of Arts, Sciences & Education at FIU Digital Commons. It has been accepted for inclusion in School of Education and Human Development Faculty Publications by an authorized administrator of FIU Digital Commons. For more information, please contact [email protected]. CINDY STENGER, KIRK WELLER, ILANA ARNON, ED DUBINSKY, DRAGA VIDAKOVIC A SEARCH FOR A CONSTRUCTIVIST APPROACH FOR UNDERSTANDING THE UNCOUNTABLE SET P(N) RESUMEN. En el presente estudio nos preguntamos si los individuos construyen estructuras mentales para el conjunto P(N) que da significado a la expresión “todos los subconjuntos de N ”. Los aportes de nuestra investigación en relación con esta pregunta tienen dos vertientes. Primeramente, identificamos las perspectivas constructivistas que han sido o podrían haber sido utilizadas para describir los mecanismos de pensamiento acerca de los conjuntos infinitos, en particular el conjunto de los números naturales.
    [Show full text]
  • On the Necessary Use of Abstract Set Theory
    ADVANCES IN MATHEMATICS 41, 209-280 (1981) On the Necessary Use of Abstract Set Theory HARVEY FRIEDMAN* Department of Mathematics, Ohio State University, Columbus, Ohio 43210 In this paper we present some independence results from the Zermelo-Frankel axioms of set theory with the axiom of choice (ZFC) which differ from earlier such independence results in three major respects. Firstly, these new propositions that are shown to be independent of ZFC (i.e., neither provable nor refutable from ZFC) form mathematically natural assertions about Bore1 functions of several variables from the Hilbert cube I” into the unit interval, or back into the Hilbert cube. As such, they are of a level of abstraction significantly below that of the earlier independence results. Secondly, these propositions are not only independent of ZFC, but also of ZFC together with the axiom of constructibility (V = L). The only earlier examples of intelligible statements independent of ZFC + V= L either express properties of formal systems such as ZFC (e.g., the consistency of ZFC), or assert the existence of very large cardinalities (e.g., inaccessible cardinals). The great bulk of independence results from ZFCLthe ones that involve standard mathematical concepts and constructions-are about sets of limited cardinality (most commonly, that of at most the continuum), and are obtained using the forcing method introduced by Paul J. Cohen (see [2]). It is now known in virtually every such case, that these independence results are eliminated if V= L is added to ZFC. Finally, some of our propositions can be proved in the theory of classes, as formalized by the Morse-Kelley class theory with the axiom of choice for sets (MKC), but not in ZFC.
    [Show full text]
  • On Cantor's Uncountability Proofs
    On Cantor's important proofs W. Mueckenheim University of Applied Sciences, Baumgartnerstrasse 16, D-86161 Augsburg, Germany [email protected] ________________________________________________________ Abstract. It is shown that the pillars of transfinite set theory, namely the uncountability proofs, do not hold. (1) Cantor's first proof of the uncountability of the set of all real numbers does not apply to the set of irrational numbers alone, and, therefore, as it stands, supplies no distinction between the uncountable set of irrational numbers and the countable set of rational numbers. (2) As Cantor's second uncountability proof, his famous second diagonalization method, is an impossibility proof, a simple counter-example suffices to prove its failure. (3) The contradiction of any bijection between a set and its power set is a consequence of the impredicative definition involved. (4) In an appendix it is shown, by a less important proof of Cantor, how transfinite set theory can veil simple structures. 1. Introduction Two finite sets have the same cardinality if there exists a one-to-one correspondence or bijection between them. Cantor extrapolated this theorem to include infinite sets as well: If between a set M and the set Ù of all natural numbers a bijection M ¨ Ù can be established, then M is denumerable or 1 countable and it has the same cardinality as Ù, namely ¡0. There may be many non-bijective mappings, but at least one bijective mapping must exist. In 1874 Cantor [1] published the proof that the set ¿ of all algebraic numbers (including the set – of all rational numbers) is denumerable.
    [Show full text]
  • Paul B. Larson a BRIEF HISTORY of DETERMINACY §1. Introduction
    Paul B. Larson A BRIEF HISTORY OF DETERMINACY x1. Introduction. Determinacy axioms are statements to the effect that certain games are determined, in that each player in the game has an optimal strategy. The commonly accepted axioms for mathematics, the Zermelo-Fraenkel axioms with the Axiom of Choice (ZFC; see [??, ??]), im- ply the determinacy of many games that people actually play. This applies in particular to many games of perfect information, games in which the players alternate moves which are known to both players, and the out- come of the game depends only on this list of moves, and not on chance or other external factors. Games of perfect information which must end in finitely many moves are determined. This follows from the work of Ernst Zermelo [??], D´enesK}onig[??] and L´aszl´oK´almar[??], and also from the independent work of John von Neumann and Oskar Morgenstern (in their 1944 book, reprinted as [??]). As pointed out by Stanis law Ulam [??], determinacy for games of perfect information of a fixed finite length is essentially a theorem of logic. If we let x1,y1,x2,y2,::: ,xn,yn be variables standing for the moves made by players player I (who plays x1,::: ,xn) and player II (who plays y1,::: ,yn), and A (consisting of sequences of length 2n) is the set of runs of the game for which player I wins, the statement (1) 9x18y1 ::: 9xn8ynhx1; y1; : : : ; xn; yni 2 A essentially asserts that the first player has a winning strategy in the game, and its negation, (2) 8x19y1 ::: 8xn9ynhx1; y1; : : : ; xn; yni 62 A essentially asserts that the second player has a winning strategy.1 We let ! denote the set of natural numbers 0; 1; 2;::: ; for brevity we will often refer to the members of this set as \integers".
    [Show full text]
  • SET THEORY Andrea K. Dieterly a Thesis Submitted to the Graduate
    SET THEORY Andrea K. Dieterly A Thesis Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of MASTER OF ARTS August 2011 Committee: Warren Wm. McGovern, Advisor Juan Bes Rieuwert Blok i Abstract Warren Wm. McGovern, Advisor This manuscript was to show the equivalency of the Axiom of Choice, Zorn's Lemma and Zermelo's Well-Ordering Principle. Starting with a brief history of the development of set history, this work introduced the Axioms of Zermelo-Fraenkel, common applications of the axioms, and set theoretic descriptions of sets of numbers. The book, Introduction to Set Theory, by Karel Hrbacek and Thomas Jech was the primary resource with other sources providing additional background information. ii Acknowledgements I would like to thank Warren Wm. McGovern for his assistance and guidance while working and writing this thesis. I also want to thank Reiuwert Blok and Juan Bes for being on my committee. Thank you to Dan Shifflet and Nate Iverson for help with the typesetting program LATEX. A personal thank you to my husband, Don, for his love and support. iii Contents Contents . iii 1 Introduction 1 1.1 Naive Set Theory . 2 1.2 The Axiom of Choice . 4 1.3 Russell's Paradox . 5 2 Axioms of Zermelo-Fraenkel 7 2.1 First Order Logic . 7 2.2 The Axioms of Zermelo-Fraenkel . 8 2.3 The Recursive Theorem . 13 3 Development of Numbers 16 3.1 Natural Numbers and Integers . 16 3.2 Rational Numbers . 20 3.3 Real Numbers .
    [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]