The Theory of Finitely Supported Structures and Choice Forms 11
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A Proof of Cantor's Theorem
Cantor’s Theorem Joe Roussos 1 Preliminary ideas Two sets have the same number of elements (are equinumerous, or have the same cardinality) iff there is a bijection between the two sets. Mappings: A mapping, or function, is a rule that associates elements of one set with elements of another set. We write this f : X ! Y , f is called the function/mapping, the set X is called the domain, and Y is called the codomain. We specify what the rule is by writing f(x) = y or f : x 7! y. e.g. X = f1; 2; 3g;Y = f2; 4; 6g, the map f(x) = 2x associates each element x 2 X with the element in Y that is double it. A bijection is a mapping that is injective and surjective.1 • Injective (one-to-one): A function is injective if it takes each element of the do- main onto at most one element of the codomain. It never maps more than one element in the domain onto the same element in the codomain. Formally, if f is a function between set X and set Y , then f is injective iff 8a; b 2 X; f(a) = f(b) ! a = b • Surjective (onto): A function is surjective if it maps something onto every element of the codomain. It can map more than one thing onto the same element in the codomain, but it needs to hit everything in the codomain. Formally, if f is a function between set X and set Y , then f is surjective iff 8y 2 Y; 9x 2 X; f(x) = y Figure 1: Injective map. -
Linear Transformation (Sections 1.8, 1.9) General View: Given an Input, the Transformation Produces an Output
Linear Transformation (Sections 1.8, 1.9) General view: Given an input, the transformation produces an output. In this sense, a function is also a transformation. 1 4 3 1 3 Example. Let A = and x = 1 . Describe matrix-vector multiplication Ax 2 0 5 1 1 1 in the language of transformation. 1 4 3 1 31 5 Ax b 2 0 5 11 8 1 Vector x is transformed into vector b by left matrix multiplication Definition and terminologies. Transformation (or function or mapping) T from Rn to Rm is a rule that assigns to each vector x in Rn a vector T(x) in Rm. • Notation: T: Rn → Rm • Rn is the domain of T • Rm is the codomain of T • T(x) is the image of vector x • The set of all images T(x) is the range of T • When T(x) = Ax, A is a m×n size matrix. Range of T = Span{ column vectors of A} (HW1.8.7) See class notes for other examples. Linear Transformation --- Existence and Uniqueness Questions (Section 1.9) Definition 1: T: Rn → Rm is onto if each b in Rm is the image of at least one x in Rn. • i.e. codomain Rm = range of T • When solve T(x) = b for x (or Ax=b, A is the standard matrix), there exists at least one solution (Existence question). Definition 2: T: Rn → Rm is one-to-one if each b in Rm is the image of at most one x in Rn. • i.e. When solve T(x) = b for x (or Ax=b, A is the standard matrix), there exists either a unique solution or none at all (Uniqueness question). -
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. -
Functions and Inverses
Functions and Inverses CS 2800: Discrete Structures, Spring 2015 Sid Chaudhuri Recap: Relations and Functions ● A relation between sets A !the domain) and B !the codomain" is a set of ordered pairs (a, b) such that a ∈ A, b ∈ B !i.e. it is a subset o# A × B" Cartesian product – The relation maps each a to the corresponding b ● Neither all possible a%s, nor all possible b%s, need be covered – Can be one-one, one&'an(, man(&one, man(&man( Alice CS 2800 Bob A Carol CS 2110 B David CS 3110 Recap: Relations and Functions ● ) function is a relation that maps each element of A to a single element of B – Can be one-one or man(&one – )ll elements o# A must be covered, though not necessaril( all elements o# B – Subset o# B covered b( the #unction is its range/image Alice Balch Bob A Carol Jameson B David Mews Recap: Relations and Functions ● Instead of writing the #unction f as a set of pairs, e usually speci#y its domain and codomain as: f : A → B * and the mapping via a rule such as: f (Heads) = 0.5, f (Tails) = 0.5 or f : x ↦ x2 +he function f maps x to x2 Recap: Relations and Functions ● Instead of writing the #unction f as a set of pairs, e usually speci#y its domain and codomain as: f : A → B * and the mapping via a rule such as: f (Heads) = 0.5, f (Tails) = 0.5 2 or f : x ↦ x f(x) ● Note: the function is f, not f(x), – f(x) is the value assigned b( f the #unction f to input x x Recap: Injectivity ● ) function is injective (one-to-one) if every element in the domain has a unique i'age in the codomain – +hat is, f(x) = f(y) implies x = y Albany NY New York A MA Sacramento B CA Boston .. -
On Stone's Theorem and the Axiom of Choice
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 4, April 1998, Pages 1211{1218 S 0002-9939(98)04163-X ON STONE'S THEOREM AND THE AXIOM OF CHOICE C. GOOD, I. J. TREE, AND W. S. WATSON (Communicated by Andreas R. Blass) Abstract. It is a well established fact that in Zermelo-Fraenkel set theory, Tychonoff's Theorem, the statement that the product of compact topological spaces is compact, is equivalent to the Axiom of Choice. On the other hand, Urysohn's Metrization Theorem, that every regular second countable space is metrizable, is provable from just the ZF axioms alone. A. H. Stone's The- orem, that every metric space is paracompact, is considered here from this perspective. Stone's Theorem is shown not to be a theorem in ZF by a forc- ing argument. The construction also shows that Stone's Theorem cannot be proved by additionally assuming the Principle of Dependent Choice. Introduction Given an infinite set X, is it possible to define a Hausdorff topology on X such that X has at least two non-isolated points? In ZFC, the answer is easily shown to be yes. However, models of ZF exist that contain infinite sets that cannot be expressed as the union of two disjoint infinite sets (amorphous sets—see [6]). For any model of ZF containing an amorphous set, the answer is no. So, as we see, even the most innocent of topological questions may be undecidable from the Zermelo- Fraenkel axioms alone. Further examples of the counter-intuitive behaviour of ‘choiceless topology’ can be found in [3] and [4]. -
Lecture Notes: Axiomatic Set Theory
Lecture Notes: Axiomatic Set Theory Asaf Karagila Last Update: May 14, 2018 Contents 1 Introduction 3 1.1 Why do we need axioms?...............................3 1.2 Classes and sets.....................................4 1.3 The axioms of set theory................................5 2 Ordinals, recursion and induction7 2.1 Ordinals.........................................8 2.2 Transfinite induction and recursion..........................9 2.3 Transitive classes.................................... 10 3 The relative consistency of the Axiom of Foundation 12 4 Cardinals and their arithmetic 15 4.1 The definition of cardinals............................... 15 4.2 The Aleph numbers.................................. 17 4.3 Finiteness........................................ 18 5 Absoluteness and reflection 21 5.1 Absoluteness...................................... 21 5.2 Reflection........................................ 23 6 The Axiom of Choice 25 6.1 The Axiom of Choice.................................. 25 6.2 Weak version of the Axiom of Choice......................... 27 7 Sets of Ordinals 31 7.1 Cofinality........................................ 31 7.2 Some cardinal arithmetic............................... 32 7.3 Clubs and stationary sets............................... 33 7.4 The Club filter..................................... 35 8 Inner models of ZF 37 8.1 Inner models...................................... 37 8.2 Gödel’s constructible universe............................. 39 1 8.3 The properties of L ................................... 41 8.4 Ordinal definable sets................................. 42 9 Some combinatorics on ω1 43 9.1 Aronszajn trees..................................... 43 9.2 Diamond and Suslin trees............................... 44 10 Coda: Games and determinacy 46 2 Chapter 1 Introduction 1.1 Why do we need axioms? In modern mathematics, axioms are given to define an object. The axioms of a group define the notion of a group, the axioms of a Banach space define what it means for something to be a Banach space. -
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. -
Math 101 B-Packet
Math 101 B-Packet Scott Rome Winter 2012-13 1 Redefining functions This quarter we have defined a function as a rule which assigns exactly one output to each input, and so far we have been happy with this definition. Unfortunately, this way of thinking of a function is insufficient as things become more complicated in mathematics. For a better understanding of a function, we will first need to define it better. Definition 1.1. Let X; Y be any sets. A function f : X ! Y is a rule which assigns every element of X to an element of Y . The sets X and Y are called the domain and codomain of f respectively. x f(x) y Figure 1: This function f : X ! Y maps x 7! f(x). The green circle indicates the range of the function. Notice y is in the codomain, but f does not map to it. Remark 1.2. It is necessary to define the rule, the domain, and the codomain to define a function. Thus far in the class, we have been \sloppy" when working with functions. Remark 1.3. Notice how in the definition, the function is defined by three things: the rule, the domain, and the codomain. That means you can define functions that seem to be the same, but are actually different as we will see. The domain of a function can be thought of as the set of all inputs (that is, everything in the domain will be mapped somewhere by the function). On the other hand, the codomain of a function is the set of all possible outputs, and a function may not necessarily map to every element of the codomain. -
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. -
The Axiom of Choice
THE AXIOM OF CHOICE THOMAS J. JECH State University of New York at Bufalo and The Institute for Advanced Study Princeton, New Jersey 1973 NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM LONDON AMERICAN ELSEVIER PUBLISHING COMPANY, INC. - NEW YORK 0 NORTH-HOLLAND PUBLISHING COMPANY - 1973 AN Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the Copyright owner. Library of Congress Catalog Card Number 73-15535 North-Holland ISBN for the series 0 7204 2200 0 for this volume 0 1204 2215 2 American Elsevier ISBN 0 444 10484 4 Published by: North-Holland Publishing Company - Amsterdam North-Holland Publishing Company, Ltd. - London Sole distributors for the U.S.A. and Canada: American Elsevier Publishing Company, Inc. 52 Vanderbilt Avenue New York, N.Y. 10017 PRINTED IN THE NETHERLANDS To my parents PREFACE The book was written in the long Buffalo winter of 1971-72. It is an attempt to show the place of the Axiom of Choice in contemporary mathe- matics. Most of the material covered in the book deals with independence and relative strength of various weaker versions and consequences of the Axiom of Choice. Also included are some other results that I found relevant to the subject. The selection of the topics and results is fairly comprehensive, nevertheless it is a selection and as such reflects the personal taste of the author. So does the treatment of the subject. The main tool used throughout the text is Cohen’s method of forcing. -
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). -
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.