DOCSLIB.ORG

Lecture 9: Axiomatic Set Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture 9: Axiomatic Set Theory Lecture 9: Axiomatic Set Theory December 16, 2014 Lecture 9: Axiomatic Set Theory Key points of today's lecture: I The iterative concept of set. I The language of set theory (LOST). I The axioms of Zermelo-Fraenkel set theory (ZFC). I Justification of the axioms based on the iterative concept of set. Today Lecture 9: Axiomatic Set Theory I The iterative concept of set. I The language of set theory (LOST). I The axioms of Zermelo-Fraenkel set theory (ZFC). I Justification of the axioms based on the iterative concept of set. Today Key points of today's lecture: Lecture 9: Axiomatic Set Theory I The language of set theory (LOST). I The axioms of Zermelo-Fraenkel set theory (ZFC). I Justification of the axioms based on the iterative concept of set. Today Key points of today's lecture: I The iterative concept of set. Lecture 9: Axiomatic Set Theory I The axioms of Zermelo-Fraenkel set theory (ZFC). I Justification of the axioms based on the iterative concept of set. Today Key points of today's lecture: I The iterative concept of set. I The language of set theory (LOST). Lecture 9: Axiomatic Set Theory I Justification of the axioms based on the iterative concept of set. Today Key points of today's lecture: I The iterative concept of set. I The language of set theory (LOST). I The axioms of Zermelo-Fraenkel set theory (ZFC). Lecture 9: Axiomatic Set Theory Today Key points of today's lecture: I The iterative concept of set. I The language of set theory (LOST). I The axioms of Zermelo-Fraenkel set theory (ZFC). I Justification of the axioms based on the iterative concept of set. Lecture 9: Axiomatic Set Theory The language of set theory (LOST) consists of the logical symbols and =, and a single binary relation symbol, 2, called membership. Note that the atomic formulas of LOST are of the form x = y and x 2 y: We will write x 6= y and x 2= y for the negation of x = y and x 2 y, respectively. The language of set theory, I Lecture 9: Axiomatic Set Theory Note that the atomic formulas of LOST are of the form x = y and x 2 y: We will write x 6= y and x 2= y for the negation of x = y and x 2 y, respectively. The language of set theory, I The language of set theory (LOST) consists of the logical symbols and =, and a single binary relation symbol, 2, called membership. Lecture 9: Axiomatic Set Theory x = y and x 2 y: We will write x 6= y and x 2= y for the negation of x = y and x 2 y, respectively. The language of set theory, I The language of set theory (LOST) consists of the logical symbols and =, and a single binary relation symbol, 2, called membership. Note that the atomic formulas of LOST are of the form Lecture 9: Axiomatic Set Theory We will write x 6= y and x 2= y for the negation of x = y and x 2 y, respectively. The language of set theory, I The language of set theory (LOST) consists of the logical symbols and =, and a single binary relation symbol, 2, called membership. Note that the atomic formulas of LOST are of the form x = y and x 2 y: Lecture 9: Axiomatic Set Theory x 6= y and x 2= y for the negation of x = y and x 2 y, respectively. The language of set theory, I The language of set theory (LOST) consists of the logical symbols and =, and a single binary relation symbol, 2, called membership. Note that the atomic formulas of LOST are of the form x = y and x 2 y: We will write Lecture 9: Axiomatic Set Theory for the negation of x = y and x 2 y, respectively. The language of set theory, I The language of set theory (LOST) consists of the logical symbols and =, and a single binary relation symbol, 2, called membership. Note that the atomic formulas of LOST are of the form x = y and x 2 y: We will write x 6= y and x 2= y Lecture 9: Axiomatic Set Theory The language of set theory, I The language of set theory (LOST) consists of the logical symbols and =, and a single binary relation symbol, 2, called membership. Note that the atomic formulas of LOST are of the form x = y and x 2 y: We will write x 6= y and x 2= y for the negation of x = y and x 2 y, respectively. Lecture 9: Axiomatic Set Theory In addition to the variables of our formal language, v1; v2;:::, we will also use the letters x; y; z; u; w; x1; x2;:::; y1; y2;:::; w1; w2;::: to stand for arbitrary formal variables. We will allow ourselves to use the standard abbreviations: _; ^; $ and 9 for \or", \and", \if and only if", and \there exists". The language of set theory, II Lecture 9: Axiomatic Set Theory x; y; z; u; w; x1; x2;:::; y1; y2;:::; w1; w2;::: to stand for arbitrary formal variables. We will allow ourselves to use the standard abbreviations: _; ^; $ and 9 for \or", \and", \if and only if", and \there exists". The language of set theory, II In addition to the variables of our formal language, v1; v2;:::, we will also use the letters Lecture 9: Axiomatic Set Theory to stand for arbitrary formal variables. We will allow ourselves to use the standard abbreviations: _; ^; $ and 9 for \or", \and", \if and only if", and \there exists". The language of set theory, II In addition to the variables of our formal language, v1; v2;:::, we will also use the letters x; y; z; u; w; x1; x2;:::; y1; y2;:::; w1; w2;::: Lecture 9: Axiomatic Set Theory We will allow ourselves to use the standard abbreviations: _; ^; $ and 9 for \or", \and", \if and only if", and \there exists". The language of set theory, II In addition to the variables of our formal language, v1; v2;:::, we will also use the letters x; y; z; u; w; x1; x2;:::; y1; y2;:::; w1; w2;::: to stand for arbitrary formal variables. Lecture 9: Axiomatic Set Theory The language of set theory, II In addition to the variables of our formal language, v1; v2;:::, we will also use the letters x; y; z; u; w; x1; x2;:::; y1; y2;:::; w1; w2;::: to stand for arbitrary formal variables. We will allow ourselves to use the standard abbreviations: _; ^; $ and 9 for \or", \and", \if and only if", and \there exists". Lecture 9: Axiomatic Set Theory 0. Axiom of Set Existence: 9v1v1 = v1: (There is a set.) 1. Axiom of Extensionality: 8v18v2(8v3(v3 2 v1 $ v3 2 v2) ! v1 = v2): (Sets that have the same members are identical.) Axioms 0 and 1 Lecture 9: Axiomatic Set Theory 9v1v1 = v1: (There is a set.) 1. Axiom of Extensionality: 8v18v2(8v3(v3 2 v1 $ v3 2 v2) ! v1 = v2): (Sets that have the same members are identical.) Axioms 0 and 1 0. Axiom of Set Existence: Lecture 9: Axiomatic Set Theory (There is a set.) 1. Axiom of Extensionality: 8v18v2(8v3(v3 2 v1 $ v3 2 v2) ! v1 = v2): (Sets that have the same members are identical.) Axioms 0 and 1 0. Axiom of Set Existence: 9v1v1 = v1: Lecture 9: Axiomatic Set Theory 1. Axiom of Extensionality: 8v18v2(8v3(v3 2 v1 $ v3 2 v2) ! v1 = v2): (Sets that have the same members are identical.) Axioms 0 and 1 0. Axiom of Set Existence: 9v1v1 = v1: (There is a set.) Lecture 9: Axiomatic Set Theory 8v18v2(8v3(v3 2 v1 $ v3 2 v2) ! v1 = v2): (Sets that have the same members are identical.) Axioms 0 and 1 0. Axiom of Set Existence: 9v1v1 = v1: (There is a set.) 1. Axiom of Extensionality: Lecture 9: Axiomatic Set Theory (Sets that have the same members are identical.) Axioms 0 and 1 0. Axiom of Set Existence: 9v1v1 = v1: (There is a set.) 1. Axiom of Extensionality: 8v18v2(8v3(v3 2 v1 $ v3 2 v2) ! v1 = v2): Lecture 9: Axiomatic Set Theory Axioms 0 and 1 0. Axiom of Set Existence: 9v1v1 = v1: (There is a set.) 1. Axiom of Extensionality: 8v18v2(8v3(v3 2 v1 $ v3 2 v2) ! v1 = v2): (Sets that have the same members are identical.) Lecture 9: Axiomatic Set Theory 2. Axiom of Foundation: 8v1(9v2 v2 2 v1 ! 9v2(v2 2 v1 ^ 8v3 v3 2= v1 _ v3 2= v2)): (Every non-empty set has a member which has no members in common with it.) Axiom 2 Lecture 9: Axiomatic Set Theory 8v1(9v2 v2 2 v1 ! 9v2(v2 2 v1 ^ 8v3 v3 2= v1 _ v3 2= v2)): (Every non-empty set has a member which has no members in common with it.) Axiom 2 2. Axiom of Foundation: Lecture 9: Axiomatic Set Theory (Every non-empty set has a member which has no members in common with it.) Axiom 2 2. Axiom of Foundation: 8v1(9v2 v2 2 v1 ! 9v2(v2 2 v1 ^ 8v3 v3 2= v1 _ v3 2= v2)): Lecture 9: Axiomatic Set Theory Axiom 2 2. Axiom of Foundation: 8v1(9v2 v2 2 v1 ! 9v2(v2 2 v1 ^ 8v3 v3 2= v1 _ v3 2= v2)): (Every non-empty set has a member which has no members in common with it.) Lecture 9: Axiomatic Set Theory 3. Axiom Schema of Comprehension: For each LOST formula '(x; z; w1;:::; wn) with the (distinct) free variables among those shown, the following is an axiom: 8w1 · · · 8wn8z9y8x(x 2 y $ (x 2 z ^ ')): (For any sets z and any property P which can be expressed by a formula of LOST, there is a set whose members are those members of z that have property P.) ! Axiom 3 Lecture 9: Axiomatic Set Theory 8w1 · · · 8wn8z9y8x(x 2 y $ (x 2 z ^ ')): (For any sets z and any property P which can be expressed by a formula of LOST, there is a set whose members are those members of z that have property P.) ! Axiom 3 3.
Load more

Recommended publications
  • A Taste of Set Theory for Philosophers
    Journal of the Indian Council of Philosophical Research, Vol. XXVII, No. 2. A Special Issue on "Logic and Philosophy Today", 143-163, 2010. Reprinted in "Logic and Philosophy Today" (edited by A. Gupta ans J.v.Benthem), College Publications vol 29, 141-162, 2011. A taste of set theory for philosophers Jouko Va¨an¨ anen¨ ∗ Department of Mathematics and Statistics University of Helsinki and Institute for Logic, Language and Computation University of Amsterdam November 17, 2010 Contents 1 Introduction 1 2 Elementary set theory 2 3 Cardinal and ordinal numbers 3 3.1 Equipollence . 4 3.2 Countable sets . 6 3.3 Ordinals . 7 3.4 Cardinals . 8 4 Axiomatic set theory 9 5 Axiom of Choice 12 6 Independence results 13 7 Some recent work 14 7.1 Descriptive Set Theory . 14 7.2 Non well-founded set theory . 14 7.3 Constructive set theory . 15 8 Historical Remarks and Further Reading 15 ∗Research partially supported by grant 40734 of the Academy of Finland and by the EUROCORES LogICCC LINT programme. I Journal of the Indian Council of Philosophical Research, Vol. XXVII, No. 2. A Special Issue on "Logic and Philosophy Today", 143-163, 2010. Reprinted in "Logic and Philosophy Today" (edited by A. Gupta ans J.v.Benthem), College Publications vol 29, 141-162, 2011. 1 Introduction Originally set theory was a theory of infinity, an attempt to understand infinity in ex- act terms. Later it became a universal language for mathematics and an attempt to give a foundation for all of mathematics, and thereby to all sciences that are based on mathematics.
    [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]
  • Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Boise State University, [email protected]
    Boise State University ScholarWorks Mathematics Undergraduate Theses Department of Mathematics 5-2014 Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Boise State University, [email protected] Follow this and additional works at: http://scholarworks.boisestate.edu/ math_undergraduate_theses Part of the Set Theory Commons Recommended Citation Rahim, Farighon Abdul, "Axioms of Set Theory and Equivalents of Axiom of Choice" (2014). Mathematics Undergraduate Theses. Paper 1. Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Advisor: Samuel Coskey Boise State University May 2014 1 Introduction Sets are all around us. A bag of potato chips, for instance, is a set containing certain number of individual chip’s that are its elements. University is another example of a set with students as its elements. By elements, we mean members. But sets should not be confused as to what they really are. A daughter of a blacksmith is an element of a set that contains her mother, father, and her siblings. Then this set is an element of a set that contains all the other families that live in the nearby town. So a set itself can be an element of a bigger set. In mathematics, axiom is defined to be a rule or a statement that is accepted to be true regardless of having to prove it. In a sense, axioms are self evident. In set theory, we deal with sets. Each time we state an axiom, we will do so by considering sets. Example of the set containing the blacksmith family might make it seem as if sets are finite.
    [Show full text]
  • Lecture 3: Constructing the Natural Numbers 1 Building
    Math/CS 120: Intro. to Math Professor: Padraic Bartlett Lecture 3: Constructing the Natural Numbers Weeks 3-4 UCSB 2014 When we defined what a proof was in our first set of lectures, we mentioned that we wanted our proofs to only start by assuming \true" statements, which we said were either previously proven-to-be-true statements or a small handful of axioms, mathematical statements which we are assuming to be true. At the time, we \handwaved" away what those axioms were, in favor of using known properties/definitions to prove results! In this talk, however, we're going to delve into the bedrock of exactly \what" properties are needed to build up some of our favorite number systems. 1 Building the Natural Numbers 1.1 First attempts. Intuitively, we think of the natural numbers as the following set: Definition. The natural numbers, denoted as N, is the set of the positive whole numbers. We denote it as follows: N = f0; 1; 2; 3;:::g This is a fine definition for most of the mathematics we will perform in this class! However, suppose that you were feeling particularly paranoid about your fellow mathematicians; i.e. you have a sneaking suspicion that the Goldbach conjecture is false, and it somehow boils down to the natural numbers being ill-defined. Or you think that you can prove P = NP with the corollary that P = NP, and to do that you need to figure out what people mean by these blackboard-bolded letters. Or you just wanted to troll your professors in CCS.
    [Show full text]
  • Set-Theoretical Background 1.1 Ordinals and Cardinals
    Set-Theoretical Background 11 February 2019 Our set-theoretical framework will be the Zermelo{Fraenkel axioms with the axiom of choice (ZFC): • Axiom of Extensionality. If X and Y have the same elements, then X = Y . • Axiom of Pairing. For all a and b there exists a set fa; bg that contains exactly a and b. • Axiom Schema of Separation. If P is a property (with a parameter p), then for all X and p there exists a set Y = fx 2 X : P (x; p)g that contains all those x 2 X that have the property P . • Axiom of Union. For any X there exists a set Y = S X, the union of all elements of X. • Axiom of Power Set. For any X there exists a set Y = P (X), the set of all subsets of X. • Axiom of Infinity. There exists an infinite set. • Axiom Schema of Replacement. If a class F is a function, then for any X there exists a set Y = F (X) = fF (x): x 2 Xg. • Axiom of Regularity. Every nonempty set has a minimal element for the membership relation. • Axiom of Choice. Every family of nonempty sets has a choice function. 1.1 Ordinals and cardinals A set X is well ordered if it is equipped with a total order relation such that every nonempty subset S ⊆ X has a smallest element. The statement that every set admits a well ordering is equivalent to the axiom of choice. A set X is transitive if every element of an element of X is an element of X.
    [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]
  • Set Theory Handout
    Salem State College MAT 420 Set Theory L. Pedro Poitevin October 25, 2007 1 Foundations De¯nition (Informal). We de¯ne a set A to be any unordered collection of objects, e.g., f3; 8; 5; 1g is a set. If x is an object, we say that x is an element of A or x 2 A if x lies in the collection; otherwise we say that x is not an element of A and write x2 = A. For instance, 3 2 f1; 2; 3; 4; 5g but 7 2= f1; 2; 3; 4; 5g. Axiom 1 (Sets are objects). If A is a set, then A is also an object. In particular, given two sets A and B, it is meaningful to ask whether A is an element of B. De¯nition (Equality of sets). Two sets A and B are equal|in notation A = B|if and only if every element of A is an element of B and vice versa. Axiom 2 (Empty set). There exists a set ;, known as the empty set, which contains no elements, i.e. for every object x we have x2 = ;. Exercise. Prove that there is only one empty set. Lemma 1 (Single choice). Let A be a nonempty set. Then there exists an object x such that x 2 A. Axiom 3 (Singletons). If a is an object, then there exists a set fag whose only element is a. Axiom 4 (Pairwise union). Given any two sets A and B, there exists a set A [ B called the union of A and B, whose elements are all objects which are elements of A or B.
    [Show full text]
  • What Are Axioms of Set Theory?
    What are Axioms of Set Theory? Set-theorists use the term Axiom (of Set Theory) quite freely. What do they mean by it? Examples Axioms of ZFC: Axiom of Extensionality Pairing Axiom Separation Axiom Union Axiom Powerset Axiom Axiom of Innity Replacement Axiom Axiom of Foundation Axiom of Choice What are Axioms of Set Theory? Beyond ZFC: Axiom of Constructibility Axiom of Determinacy Large Cardinal Axioms Reinhardt's Axiom Cardinal Characteristic Axioms Martin's Axiom Axiom A Proper Forcing Axiom Open Colouring Axiom What are Axioms of Set Theory? Hypotheses: Continuum Hypothesis Suslin Hypothesis Kurepa Hypothesis Singular Cardinal Hypothesis Principles: Diamond Principle Square Principle Vopenka's Principle Reection Principles What are Axioms of Set Theory? When does a statement achieve the status of axiom, hypothesis or principle? It is worth examining this question in three specic cases: A. The Pairing Axiom B. Large Cardinal Axioms C. The Axiom of Choice What are Axioms of Set Theory? A. The Pairing Axiom If x; y are sets then there is a set whose elements are precisely x and y Such an assertion is basic to the way sets are regarded in mathematics. Moreover if the term set is used in a way that violates this assertion we would have to regard this use as based upon a dierent concept altogether. Thus, as Feferman has observed, the Pairing Axiom qualies as an axiom in the ideal sense of the Oxford English Dictionary: A self-evident proposition requiring no formal demonstration to prove its truth, but received and assented to as soon as mentioned.
    [Show full text]
  • Axioms and Models for an Extended Set Theory
    AXIOMS AND MODELS FOR AN EXTENDED SET THEORY ANDREAS BLASS AND D L CHILDS Abstract. We present the axioms of extended set theory (XST) and the ideas underlying the axioms. We also present an interpre- tation of XST in ZFC plus “there exist arbitrarily large inaccessible cardinals,” thereby proving the consistency of XST relative to this mild large-cardinal extension of ZFC. 1. Introduction This paper presents an extended set theory (XST) and proves its consistency relative to the classical Zermelo-Fraenkel set theory with the axiom of choice (ZFC) and an axiom asserting the existence of arbitrarily large inaccessible cardinals (also known as Grothendieck’s axiom of universes). The original motivation for this development of XST was to provide an improved, flexible mathematical foundation for certain constructions, like tuples and functions, that play important roles in computing. Subsequent developments revealed that XST also provides a smooth way of handling the set-class distinction that turns up especially in category theory. We shall comment on these matters in more detail after the axioms of XST have been introduced. The extended set theory XST differs from Zermelo-Fraenkel set the- ory ZFC in three major ways (and several minor ways). The biggest difference between XST and ZFC is that the fundamental membership relation of XST is a ternary relation, written x ∈s y and read “x is an element of y with scope s.” Most of the axioms of XST are analogs of traditional ZFC axioms, modified to account for membership scopes, and allowing atoms (also called urelements). This aspect of XST can be interpreted in ZFC quite straightforwardly.
    [Show full text]
  • The Axiom of Choice and Its Implications in Mathematics
    Treball final de grau GRAU DE MATEMATIQUES` Facultat de Matem`atiquesi Inform`atica Universitat de Barcelona The Axiom of Choice and its implications in mathematics Autor: Gina Garcia Tarrach Director: Dr. Joan Bagaria Realitzat a: Departament de Matem`atiques i Inform`atica Barcelona, 29 de juny de 2017 Abstract The Axiom of Choice is an axiom of set theory which states that, given a collection of non-empty sets, it is possible to choose an element out of each set of the collection. The implications of the acceptance of the Axiom are many, some of them essential to the de- velopment of contemporary mathematics. In this work, we give a basic presentation of the Axiom and its consequences: we study the Axiom of Choice as well as some of its equivalent forms such as the Well Ordering Theorem and Zorn's Lemma, some weaker choice principles, the implications of the Axiom in different fields of mathematics, so- me paradoxical results implied by it, and its role within the Zermelo-Fraenkel axiomatic theory. i Contents Introduction 1 0 Some preliminary notes on well-orders, ordinal and cardinal numbers 3 1 Historical background 6 2 The Axiom of Choice and its Equivalent Forms 9 2.1 The Axiom of Choice . 9 2.2 The Well Ordering Theorem . 10 2.3 Zorn's Lemma . 12 2.4 Other equivalent forms . 13 3 Weaker Forms of the Axiom of Choice 14 3.1 The Axiom of Dependent Choice . 14 3.2 The Axiom of Countable Choice . 15 3.3 The Boolean Prime Ideal Theorem .
    [Show full text]
  • Basic Set Theory
    Basic set theory Richard Pettigrew January 26, 2012 1 A set and its members • A set is a collection of objects. • The objects in a set are called its elements or members. • If x is a set and z is a member of x, then we write z 2 x. • If x is a set and z is not a member of x, then we write z 62 x. • If x and y are sets and if all members of x are also members of y, then we say that x is a subset of y. We write this x ⊆ y. Thus, we have: x ⊆ y $ 8z(z 2 x ! z 2 y) • The members of a set determine that set. That is, there are no two distinct sets with the same members. This distinguishes sets from properties, concepts, and other intensional notions. This is codified in the following axiom of set theory: Axiom 1 (Axiom of Extensionality) 8x8y[x = y $ (8z)(z 2 x $ z 2 y)] Another way of stating this is that if x ⊆ y and y ⊆ x, then x = y. This axiom is reasonably uncontroversial and it is shared by nearly all versions of set theory. 2 Which sets are there? 2.1 Na¨ıve set theory Na¨ıveSet Theory has two axioms: • The Axiom of Extensionality. • The Axiom of Unrestricted Comprehension. This says that, for any property, there is a set of all and only those things that have that property. More precisely, we restrict to properties that can be defined by formulae in the language of set theory with parameters: 1 Axiom 2 (Axiom of Unrestricted Comprehension) Given a formula Φ(x; w1; : : : ; wn) in the language of set theory, 8w1; : : : ; wn9x8z[z 2 x $ Φ(z; w1; : : : ; wn)] Given a formula Φ(x; w1; : : : ; wn) and values w1, :::, wn, we write this set fx : Φ(x; w1; : : : ; wn)g.
    [Show full text]
  • Logic and Set Theory
    Part II | Logic and Set Theory Based on lectures by I. B. Leader Notes taken by Dexter Chua Lent 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. No specific prerequisites. Ordinals and cardinals Well-orderings and order-types. Examples of countable ordinals. Uncountable ordi- nals and Hartogs' lemma. Induction and recursion for ordinals. Ordinal arithmetic. Cardinals; the hierarchy of alephs. Cardinal arithmetic. [5] Posets and Zorn's lemma Partially ordered sets; Hasse diagrams, chains, maximal elements. Lattices and Boolean algebras. Complete and chain-complete posets; fixed-point theorems. The axiom of choice and Zorn's lemma. Applications of Zorn's lemma in mathematics. The well- ordering principle. [5] Propositional logic The propositional calculus. Semantic and syntactic entailment. The deduction and completeness theorems. Applications: compactness and decidability. [3] Predicate logic The predicate calculus with equality. Examples of first-order languages and theories. Statement of the completeness theorem; *sketch of proof*. The compactness theorem and the Lowenheim-Skolem theorems. Limitations of first-order logic. Model theory. [5] Set theory Set theory as a first-order theory; the axioms of ZF set theory. Transitive closures, epsilon-induction and epsilon-recursion. Well-founded relations. Mostowski's collapsing theorem. The rank function and the von Neumann hierarchy. [5] Consistency *Problems of consistency and independence* [1] 1 Contents II Logic and Set Theory Contents 0 Introduction 3 1 Propositional calculus 4 1.1 Propositions . .4 1.2 Semantic entailment .
    [Show full text]