Elements of Set Theory by Sidney Felder in the Words of Georg Cantor
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1 Elementary Set Theory
1 Elementary Set Theory Notation: fg enclose a set. f1; 2; 3g = f3; 2; 2; 1; 3g because a set is not defined by order or multiplicity. f0; 2; 4;:::g = fxjx is an even natural numberg because two ways of writing a set are equivalent. ; is the empty set. x 2 A denotes x is an element of A. N = f0; 1; 2;:::g are the natural numbers. Z = f:::; −2; −1; 0; 1; 2;:::g are the integers. m Q = f n jm; n 2 Z and n 6= 0g are the rational numbers. R are the real numbers. Axiom 1.1. Axiom of Extensionality Let A; B be sets. If (8x)x 2 A iff x 2 B then A = B. Definition 1.1 (Subset). Let A; B be sets. Then A is a subset of B, written A ⊆ B iff (8x) if x 2 A then x 2 B. Theorem 1.1. If A ⊆ B and B ⊆ A then A = B. Proof. Let x be arbitrary. Because A ⊆ B if x 2 A then x 2 B Because B ⊆ A if x 2 B then x 2 A Hence, x 2 A iff x 2 B, thus A = B. Definition 1.2 (Union). Let A; B be sets. The Union A [ B of A and B is defined by x 2 A [ B if x 2 A or x 2 B. Theorem 1.2. A [ (B [ C) = (A [ B) [ C Proof. Let x be arbitrary. x 2 A [ (B [ C) iff x 2 A or x 2 B [ C iff x 2 A or (x 2 B or x 2 C) iff x 2 A or x 2 B or x 2 C iff (x 2 A or x 2 B) or x 2 C iff x 2 A [ B or x 2 C iff x 2 (A [ B) [ C Definition 1.3 (Intersection). -
Determinacy in Linear Rational Expectations Models
Journal of Mathematical Economics 40 (2004) 815–830 Determinacy in linear rational expectations models Stéphane Gauthier∗ CREST, Laboratoire de Macroéconomie (Timbre J-360), 15 bd Gabriel Péri, 92245 Malakoff Cedex, France Received 15 February 2002; received in revised form 5 June 2003; accepted 17 July 2003 Available online 21 January 2004 Abstract The purpose of this paper is to assess the relevance of rational expectations solutions to the class of linear univariate models where both the number of leads in expectations and the number of lags in predetermined variables are arbitrary. It recommends to rule out all the solutions that would fail to be locally unique, or equivalently, locally determinate. So far, this determinacy criterion has been applied to particular solutions, in general some steady state or periodic cycle. However solutions to linear models with rational expectations typically do not conform to such simple dynamic patterns but express instead the current state of the economic system as a linear difference equation of lagged states. The innovation of this paper is to apply the determinacy criterion to the sets of coefficients of these linear difference equations. Its main result shows that only one set of such coefficients, or the corresponding solution, is locally determinate. This solution is commonly referred to as the fundamental one in the literature. In particular, in the saddle point configuration, it coincides with the saddle stable (pure forward) equilibrium trajectory. © 2004 Published by Elsevier B.V. JEL classification: C32; E32 Keywords: Rational expectations; Selection; Determinacy; Saddle point property 1. Introduction The rational expectations hypothesis is commonly justified by the fact that individual forecasts are based on the relevant theory of the economic system. -
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. -
Infinite Sets
“mcs-ftl” — 2010/9/8 — 0:40 — page 379 — #385 13 Infinite Sets So you might be wondering how much is there to say about an infinite set other than, well, it has an infinite number of elements. Of course, an infinite set does have an infinite number of elements, but it turns out that not all infinite sets have the same size—some are bigger than others! And, understanding infinity is not as easy as you might think. Some of the toughest questions in mathematics involve infinite sets. Why should you care? Indeed, isn’t computer science only about finite sets? Not exactly. For example, we deal with the set of natural numbers N all the time and it is an infinite set. In fact, that is why we have induction: to reason about predicates over N. Infinite sets are also important in Part IV of the text when we talk about random variables over potentially infinite sample spaces. So sit back and open your mind for a few moments while we take a very brief look at infinity. 13.1 Injections, Surjections, and Bijections We know from Theorem 7.2.1 that if there is an injection or surjection between two finite sets, then we can say something about the relative sizes of the two sets. The same is true for infinite sets. In fact, relations are the primary tool for determining the relative size of infinite sets. Definition 13.1.1. Given any two sets A and B, we say that A surj B iff there is a surjection from A to B, A inj B iff there is an injection from A to B, A bij B iff there is a bijection between A and B, and A strict B iff there is a surjection from A to B but there is no bijection from B to A. -
PROBLEM SET 1. the AXIOM of FOUNDATION Early on in the Book
PROBLEM SET 1. THE AXIOM OF FOUNDATION Early on in the book (page 6) it is indicated that throughout the formal development ‘set’ is going to mean ‘pure set’, or set whose elements, elements of elements, and so on, are all sets and not items of any other kind such as chairs or tables. This convention applies also to these problems. 1. A set y is called an epsilon-minimal element of a set x if y Î x, but there is no z Î x such that z Î y, or equivalently x Ç y = Ø. The axiom of foundation, also called the axiom of regularity, asserts that any set that has any element at all (any nonempty set) has an epsilon-minimal element. Show that this axiom implies the following: (a) There is no set x such that x Î x. (b) There are no sets x and y such that x Î y and y Î x. (c) There are no sets x and y and z such that x Î y and y Î z and z Î x. 2. In the book the axiom of foundation or regularity is considered only in a late chapter, and until that point no use of it is made of it in proofs. But some results earlier in the book become significantly easier to prove if one does use it. Show, for example, how to use it to give an easy proof of the existence for any sets x and y of a set x* such that x* and y are disjoint (have empty intersection) and there is a bijection (one-to-one onto function) from x to x*, a result called the exchange principle. -
Enumeration of Finite Automata 1 Z(A) = 1
INFOI~MATION AND CONTROL 10, 499-508 (1967) Enumeration of Finite Automata 1 FRANK HARARY AND ED PALMER Department of Mathematics, University of Michigan, Ann Arbor, Michigan Harary ( 1960, 1964), in a survey of 27 unsolved problems in graphical enumeration, asked for the number of different finite automata. Re- cently, Harrison (1965) solved this problem, but without considering automata with initial and final states. With the aid of the Power Group Enumeration Theorem (Harary and Palmer, 1965, 1966) the entire problem can be handled routinely. The method involves a confrontation of several different operations on permutation groups. To set the stage, we enumerate ordered pairs of functions with respect to the product of two power groups. Finite automata are then concisely defined as certain ordered pah's of functions. We review the enumeration of automata in the natural setting of the power group, and then extend this result to enumerate automata with initial and terminal states. I. ENUMERATION THEOREM For completeness we require a number of definitions, which are now given. Let A be a permutation group of order m = ]A I and degree d acting on the set X = Ix1, x~, -.. , xa}. The cycle index of A, denoted Z(A), is defined as follows. Let jk(a) be the number of cycles of length k in the disjoint cycle decomposition of any permutation a in A. Let al, a2, ... , aa be variables. Then the cycle index, which is a poly- nomial in the variables a~, is given by d Z(A) = 1_ ~ H ~,~(°~ . (1) ~$ a EA k=l We sometimes write Z(A; al, as, .. -
The Axiom of Choice and Its Implications
THE AXIOM OF CHOICE AND ITS IMPLICATIONS KEVIN BARNUM Abstract. In this paper we will look at the Axiom of Choice and some of the various implications it has. These implications include a number of equivalent statements, and also some less accepted ideas. The proofs discussed will give us an idea of why the Axiom of Choice is so powerful, but also so controversial. Contents 1. Introduction 1 2. The Axiom of Choice and Its Equivalents 1 2.1. The Axiom of Choice and its Well-known Equivalents 1 2.2. Some Other Less Well-known Equivalents of the Axiom of Choice 3 3. Applications of the Axiom of Choice 5 3.1. Equivalence Between The Axiom of Choice and the Claim that Every Vector Space has a Basis 5 3.2. Some More Applications of the Axiom of Choice 6 4. Controversial Results 10 Acknowledgments 11 References 11 1. Introduction The Axiom of Choice states that for any family of nonempty disjoint sets, there exists a set that consists of exactly one element from each element of the family. It seems strange at first that such an innocuous sounding idea can be so powerful and controversial, but it certainly is both. To understand why, we will start by looking at some statements that are equivalent to the axiom of choice. Many of these equivalences are very useful, and we devote much time to one, namely, that every vector space has a basis. We go on from there to see a few more applications of the Axiom of Choice and its equivalents, and finish by looking at some of the reasons why the Axiom of Choice is so controversial. -
Formal Construction of a Set Theory in Coq
Saarland University Faculty of Natural Sciences and Technology I Department of Computer Science Masters Thesis Formal Construction of a Set Theory in Coq submitted by Jonas Kaiser on November 23, 2012 Supervisor Prof. Dr. Gert Smolka Advisor Dr. Chad E. Brown Reviewers Prof. Dr. Gert Smolka Dr. Chad E. Brown Eidesstattliche Erklarung¨ Ich erklare¨ hiermit an Eides Statt, dass ich die vorliegende Arbeit selbststandig¨ verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel verwendet habe. Statement in Lieu of an Oath I hereby confirm that I have written this thesis on my own and that I have not used any other media or materials than the ones referred to in this thesis. Einverstandniserkl¨ arung¨ Ich bin damit einverstanden, dass meine (bestandene) Arbeit in beiden Versionen in die Bibliothek der Informatik aufgenommen und damit vero¨ffentlicht wird. Declaration of Consent I agree to make both versions of my thesis (with a passing grade) accessible to the public by having them added to the library of the Computer Science Department. Saarbrucken,¨ (Datum/Date) (Unterschrift/Signature) iii Acknowledgements First of all I would like to express my sincerest gratitude towards my advisor, Chad Brown, who supported me throughout this work. His extensive knowledge and insights opened my eyes to the beauty of axiomatic set theory and foundational mathematics. We spent many hours discussing the minute details of the various constructions and he taught me the importance of mathematical rigour. Equally important was the support of my supervisor, Prof. Smolka, who first introduced me to the topic and was there whenever a question arose. -
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. -
Indiscernible Pairs of Countable Sets of Reals at a Given Projective Level
Indiscernible pairs of countable sets of reals at a given projective level Vladimir Kanovei∗ Vassily Lyubetsky† January 1, 2020 Abstract 1 Using an invariant modification of Jensen’s “minimal Π2 singleton” forcing, we define a model of ZFC, in which, for a given n ≥ 2, there exists 1 an Πn unordered pair of non-OD (hence, OD-indiscernible) countable sets 1 of reals, but there is no Σn unordered pairs of this kind. Any two reals x1 6= x2 are discernible by a simple formula ϕ(x) := x < r for a suitable rational r. Therefore, the lowest (type-theoretic) level of sets where one may hope to find indiscernible elements, is the level of sets of reals. And in- deed, identifying the informal notion of definability with the ordinal definability (OD), one finds indiscernible sets of reals in appropriate generic models. Example 1. If reals a 6= b in 2ω form a Cohen-generic pair over L, then ω the constructibility degrees [a]L = {x ∈ 2 : L[x] = L[a]} and [b]L are OD- indiscernible disjoint sets of reals in L[a, b], by rather straightforward forcing arguments, see [2, Theorem 3.1] and a similar argument in [3, Theorem 2.5]. Example 2. As observed in [5], if reals a 6= b in 2ω form a Sacks-generic pair over L, then the constructibility degrees [a]L and [b]L still are OD-indiscernible disjoint sets in L[a, b], with the additional advantage that the unordered pair arXiv:1912.12962v1 [math.LO] 30 Dec 2019 {[a]L, [b]L } is an OD set in L[a, b] because [a]L, [b]L are the only two minimal degrees in L[a, b]. -
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. -
Functions and Infinite Sets
Indian Institute of Information Technology Design and Manufacturing, Kancheepuram Instructor Chennai 600 127, India N.Sadagopan An Autonomous Institute under MHRD, Govt of India Scribe: An Institute of National Importance S.Dhanalakshmi www.iiitdm.ac.in A. Mohanapriya COM205T Discrete Structures for Computing-Lecture Notes Functions and Infinite Sets Objective: We shall introduce functions, special functions and related counting problems. We shall see the importance of functions in the context of infinite sets and discuss infinite sets in detail. Further, we introduce the notion counting in the context of infinite sets. Revisit: Relations Consider the following relations and its graphical representation, and this representation is dif- ferent from the one we discussed in the context of relations. A = B = f1; 2; 3; 4; 5g, R1 = f(1; 2); (1; 3); (2; 3); (2; 4); (2; 5)g A = B = f1; 2; 3g, R2 = f(1; 1); (2; 2); (3; 3)g A = B = f1; 2; 3g, R3 = f(1; 1); (2; 2); (2; 3)g A = B = f1; 2; 3g, R4 = f(1; 1); (2; 1); (3; 1)g We define special relations; relations satisfying the following properties. 1 1 1 1 1 1 1 1 2 2 3 3 2 2 2 2 2 2 4 4 5 5 3 3 3 3 3 A B A B A B A B (a) R1 (b) R2 (c) R3 (d) R4 Fig. 1: (a),(b),(c),(d) represent relations using two partitions. 1. Property P1: 8a 2 A 9!b 2 B [(a; b) 2 R] 2. Property P2: relation satisfying P1 and @a 2 A @b 2 A ^ a 6= b ^ 8c 2 B [(a; c) 2 R ^ (b; c) 2 R] 3.