DOCSLIB.ORG

The Axiom of Choice

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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. I tried to use as much similarity between sym- metric models of ZF and permutation models of ZFA as possible. The relation between these two methods is described in Chapter 6, where I pre- sent the Embedding Theorem (Jech-Sochor) and the Support Theorem (Pincus). Consistency and independence of the Axiom of Choice (due to Godel and Cohen, respectively) are presented in Chapters 3 and 5. Chapter 4 introduces the permutation models (Fraenkel, Mostowski, Specker). Chapters 2 and 10 are devoted to the use of the Axiom of Choice in mathematics. Chapter 2 contains examples of proofs using the Axiom of Choice, whereas Chapter 10 provides counterexamples in the absence of the Axiom of Choice. Chapters 7 and 8 deal with various consequences of the Axiom of Choice. The topics considered in Chapter 7 include the Prime Ideal Theorem, the Ordering Principle and the Axiom of Choice for Finite Sets, while Chapter 8 is concerned with the Countable Axiom of Choice, the Principle of De- pendent Choices and related topics. In Chapter 9 we discuss the differences between permutation models of ZFA and symmetric models of ZF. Chapter 11 is devoted to questions concerning cardinal numbers in set theory without the Axiom of Choice. Finally, in Chapter 12 we discuss VII VIII PREFACE briefly the Axiom of Determinacy and related topics. This subject has been recently in the focus of attention, and it seems that solution of the open problems presented at the end of Chapter 12 would mean an important con- tribution to the investigation of the Foundations of Mathematics. The book is more or less self-contained. Most of the theorems are accom- panied by a proof of a sort. There are, however, several instances where I elected to present a theorem without proof, and an insatiable reader will have to resort to looking up the proof in the journals. Each chapter contains a number of problems. Some of them are simply exercises, but many of the problems are actually important results which for various reasons are not included in the main text. These are in most cases provided with an outline of proof. To help the reader, I adopted the classification used by producers of French brandy: The one star problems are generally more difficult than the plain problems. Two stars are reserved for results presented without a proof (in most cases, such a result is the main theorem of a long paper). And finally, three stars indicate that the problem is open (there are still a few left). Among those whose work on the subject has had most influence on this book are (apart from Godel and Cohen) A. Levy, A. Mostowski and A. Tarski, to name a few. My special thanks are due to P. VopEnka, who is responsible for my interest in the subject and under whose guidance I wrote my 1966 thesis ‘The Axiom of Choice’. I am very much indebted to D. Pincus, for two reasons. Firstly, many of the results of his 1969 thesis found its way into this book, and secondly, his correspondence with me throughout the past five years had a significant influence on my treatment of the subject. Finally, I am grateful to U. Felgner for his valuable comments after he was so kind to read my manuscript last summer. Princeton, N.J. December 1972 CHAPTER 1 INTRODUCTION 1.1. The Axiom of Choice A mathematician of the present generation hardly considers the use of the Axiom of Choice a questionable method of proof. As a result of algebra and analysis going abstract and the development of new mathematical Is- ciplines such as set theory and topology, practically every mathematician learns about the Axiom of Choice (or at least of its most popular form, Zorn’s Lemma) in an undergraduate course. He also probably vaguely knows that there has been some controversy involving the Axiom of Choice, but it has been resolved by the logicians to a general satisfaction. There has been some controversy about the Axiom of Choice, indeed. First, let us see what the Axiom of Choice says: AXIOMOF CHOICE.For every family 9 of nonempty sets, there is u function f such that f (S)E S for each set S in the family 9. (We say that f is a choice function on F.)In some particular instances, this is self-evident. E.g., let 9 consist of sets of the form {a, b}, where a and b are real numbers. Then the function IS a choice function on 9.Or, if all the sets in 9are singletons, i.e. sets of the form {a}, then one can easily find a choice function on 9.Finally, if .F is ajkite family of nonempty sets, then there is a choice function. (To show this, one uses induction on the size of 9;naturally, a choice function exists for a family which consists of a single nonempty set.) 1 2 INTRODUCTION [CH. 1, $2 Thus the idea of having a choice function on every family of nonempty sets is not against our mathematical mind. But try to think of a choice func- tion on the family of all nonempty sets of reals! Or, let 2F be the family of all (unordered) pairs { S, T},where S and T are sets of reals. After thinking about these problems for a while, one should start to appreciate the other point of view. The Axiom of Choice is different from the ordinary principles accepted by mathematicians. And this was one of the sources of objections to the Axiom of Choice, as late as in the thirties. The other source of objections is the fact that the Axiom of Choice can be used to prove theorems which are to a certain extent ‘unpleasant’, and even theorems which are not exactly in line with our ‘common sense’ intuition. The rest of Chapter 1 is devoted to such examples. 1.2. A nonmeasurable set of real numbers The most popular example of an ‘unpleasant’ consequence of the Axiom of Choice is the existence of a set of real numbers which is not Lebesgue measurable. Let p(X) denote the Lebesgue measure of a set X of real numbers. As is well-known, p is countably additive, translation invariant and ,u( [a, b]) = b-a for every interval [a, b]. For real numbers in the interval [0, 11, let us define x-Y whenever x-y is a rational number. The relation - on [0, I] is an equiva- lence; denote by [XI the corresponding equivalence class, for each x E [0, I]. By the Axiom of Choice, we can choose one element out of each equivalence class. Thus there is a set M of real numbers, M c [0, 11, with the property that for each real number x, there exists a unique y E M and a unique rational number r such that x = y + r. If we let M,. = {y+r : y E M} for each rational number Y, then we have a partition of the set R of all real numbers into countably many disjoint sets (1.1) R = u {M,. : r rational). It follows that the set M is not measurable. Assuming that M is measurable, we get a contradiction as follows: First, p(M) = 0 is impossible, since this CH.1, $31 A PARADOXICAL DECOMPOSITION OF THE SPHERE 3 would imply p(R) = 0, using (1.1). However, p(M)> 0 is also impossible, since this would imply p([O, 11) > p(u {M, : r rational and 0 < r < l}) r rational using the fact that each Mr would have to have the same measure as M. 1.3. A paradoxical decomposition of the sphere The theorems presented here demonstrate what 'absurd' results can be expected when employing the Axiom of Choice. THEOREM1.1. A sphere S can be decomposed into disjoint sets S=AUBUCUQ such that: (i) the sets A, B, C are congruent to each other; (ii) the set B u C is congruent to each of the sets A, By C; (iii) Q is countable.
Load more

Recommended publications
  • Lecture Slides
    Program Verification: Lecture 3 Jos´eMeseguer Computer Science Department University of Illinois at Urbana-Champaign 1 Algebras An (unsorted, many-sorted, or order-sorted) signature Σ is just syntax: provides the symbols for a language; but what is that language talking about? what is its semantics? It is obviously talking about algebras, which are the mathematical models in which we interpret the syntax of Σ, giving it concrete meaning. Unsorted algebras are the simplest example: children become familiar with them from the early awakenings of reason. They consist of a set of data elements, and various chosen constants among those elements, and operations on such data. 2 Algebras (II) For example, for Σ the unsorted signature of the module NAT-MIXFIX we can define many different algebras, such as the following: 1. IN, the algebra of natural numbers in whatever notation we wish (Peano, binary, base 10, etc.) with 0 interpreted as the zero element, s interpreted as successor, and + and * interpreted as natural number addition and multiplication. 2. INk, the algebra of residue classes modulo k, for k a nonzero natural number. This is a finite algebra whose set of elements can be represented as the set {0,...,k − 1}. We interpret 0 as 0, and for the other 3 operations we perform them in IN and then take the residue modulo k. For example, in IN7 we have 6+6=5. 3. Z, the algebra of the integers, with 0 interpreted as the zero element, s interpreted as successor, and + and * interpreted as integer addition and multiplication. 4.
    [Show full text]
  • Redalyc.Sets and Pluralities
    Red de Revistas Científicas de América Latina, el Caribe, España y Portugal Sistema de Información Científica Gustavo Fernández Díez Sets and Pluralities Revista Colombiana de Filosofía de la Ciencia, vol. IX, núm. 19, 2009, pp. 5-22, Universidad El Bosque Colombia Available in: http://www.redalyc.org/articulo.oa?id=41418349001 Revista Colombiana de Filosofía de la Ciencia, ISSN (Printed Version): 0124-4620 [email protected] Universidad El Bosque Colombia How to cite Complete issue More information about this article Journal's homepage www.redalyc.org Non-Profit Academic Project, developed under the Open Acces Initiative Sets and Pluralities1 Gustavo Fernández Díez2 Resumen En este artículo estudio el trasfondo filosófico del sistema de lógica conocido como “lógica plural”, o “lógica de cuantificadores plurales”, de aparición relativamente reciente (y en alza notable en los últimos años). En particular, comparo la noción de “conjunto” emanada de la teoría axiomática de conjuntos, con la noción de “plura- lidad” que se encuentra detrás de este nuevo sistema. Mi conclusión es que los dos son completamente diferentes en su alcance y sus límites, y que la diferencia proviene de las diferentes motivaciones que han dado lugar a cada uno. Mientras que la teoría de conjuntos es una teoría genuinamente matemática, que tiene el interés matemático como ingrediente principal, la lógica plural ha aparecido como respuesta a considera- ciones lingüísticas, relacionadas con la estructura lógica de los enunciados plurales del inglés y el resto de los lenguajes naturales. Palabras clave: conjunto, teoría de conjuntos, pluralidad, cuantificación plural, lógica plural. Abstract In this paper I study the philosophical background of the relatively recent (and in the last few years increasingly flourishing) system of logic called “plural logic”, or “logic of plural quantifiers”.
    [Show full text]
  • The Iterative Conception of Set Author(S): George Boolos Reviewed Work(S): Source: the Journal of Philosophy, Vol
    Journal of Philosophy, Inc. The Iterative Conception of Set Author(s): George Boolos Reviewed work(s): Source: The Journal of Philosophy, Vol. 68, No. 8, Philosophy of Logic and Mathematics (Apr. 22, 1971), pp. 215-231 Published by: Journal of Philosophy, Inc. Stable URL: http://www.jstor.org/stable/2025204 . Accessed: 12/01/2013 10:53 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Journal of Philosophy, Inc. is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Philosophy. http://www.jstor.org This content downloaded on Sat, 12 Jan 2013 10:53:17 AM All use subject to JSTOR Terms and Conditions THE JOURNALOF PHILOSOPHY VOLUME LXVIII, NO. 8, APRIL 22, I97I _ ~ ~~~~~~- ~ '- ' THE ITERATIVE CONCEPTION OF SET A SET, accordingto Cantor,is "any collection.., intoa whole of definite,well-distinguished objects... of our intuitionor thought.'1Cantor alo sdefineda set as a "many, whichcan be thoughtof as one, i.e., a totalityof definiteelements that can be combinedinto a whole by a law.'2 One mightobject to the firstdefi- nitionon the groundsthat it uses the conceptsof collectionand whole, which are notionsno betterunderstood than that of set,that there ought to be sets of objects that are not objects of our thought,that 'intuition'is a termladen with a theoryof knowledgethat no one should believe, that any object is "definite,"that there should be sets of ill-distinguishedobjects, such as waves and trains,etc., etc.
    [Show full text]
  • Orbits of Automorphism Groups of Fields
    ORBITS OF AUTOMORPHISM GROUPS OF FIELDS KIRAN S. KEDLAYA AND BJORN POONEN Abstract. We address several specific aspects of the following general question: can a field K have so many automorphisms that the action of the automorphism group on the elements of K has relatively few orbits? We prove that any field which has only finitely many orbits under its automorphism group is finite. We extend the techniques of that proof to approach a broader conjecture, which asks whether the automorphism group of one field over a subfield can have only finitely many orbits on the complement of the subfield. Finally, we apply similar methods to analyze the field of Mal'cev-Neumann \generalized power series" over a base field; these form near-counterexamples to our conjecture when the base field has characteristic zero, but often fall surprisingly far short in positive characteristic. Can an infinite field K have so many automorphisms that the action of the automorphism group on the elements of K has only finitely many orbits? In Section 1, we prove that the answer is \no" (Theorem 1.1), even though the corresponding answer for division rings is \yes" (see Remark 1.2). Our proof constructs a \trace map" from the given field to a finite field, and exploits the peculiar combination of additive and multiplicative properties of this map. Section 2 attempts to prove a relative version of Theorem 1.1, by considering, for a non- trivial extension of fields k ⊂ K, the action of Aut(K=k) on K. In this situation each element of k forms an orbit, so we study only the orbits of Aut(K=k) on K − k.
    [Show full text]
  • The Metamathematics of Putnam's Model-Theoretic Arguments
    The Metamathematics of Putnam's Model-Theoretic Arguments Tim Button Abstract. Putnam famously attempted to use model theory to draw metaphysical conclusions. His Skolemisation argument sought to show metaphysical realists that their favourite theories have countable models. His permutation argument sought to show that they have permuted mod- els. His constructivisation argument sought to show that any empirical evidence is compatible with the Axiom of Constructibility. Here, I exam- ine the metamathematics of all three model-theoretic arguments, and I argue against Bays (2001, 2007) that Putnam is largely immune to meta- mathematical challenges. Copyright notice. This paper is due to appear in Erkenntnis. This is a pre-print, and may be subject to minor changes. The authoritative version should be obtained from Erkenntnis, once it has been published. Hilary Putnam famously attempted to use model theory to draw metaphys- ical conclusions. Specifically, he attacked metaphysical realism, a position characterised by the following credo: [T]he world consists of a fixed totality of mind-independent objects. (Putnam 1981, p. 49; cf. 1978, p. 125). Truth involves some sort of correspondence relation between words or thought-signs and external things and sets of things. (1981, p. 49; cf. 1989, p. 214) [W]hat is epistemically most justifiable to believe may nonetheless be false. (1980, p. 473; cf. 1978, p. 125) To sum up these claims, Putnam characterised metaphysical realism as an \externalist perspective" whose \favorite point of view is a God's Eye point of view" (1981, p. 49). Putnam sought to show that this externalist perspective is deeply untenable. To this end, he treated correspondence in terms of model-theoretic satisfaction.
    [Show full text]
  • Set-Theoretic Geology, the Ultimate Inner Model, and New Axioms
    Set-theoretic Geology, the Ultimate Inner Model, and New Axioms Justin William Henry Cavitt (860) 949-5686 [email protected] Advisor: W. Hugh Woodin Harvard University March 20, 2017 Submitted in partial fulfillment of the requirements for the degree of Bachelor of Arts in Mathematics and Philosophy Contents 1 Introduction 2 1.1 Author’s Note . .4 1.2 Acknowledgements . .4 2 The Independence Problem 5 2.1 Gödelian Independence and Consistency Strength . .5 2.2 Forcing and Natural Independence . .7 2.2.1 Basics of Forcing . .8 2.2.2 Forcing Facts . 11 2.2.3 The Space of All Forcing Extensions: The Generic Multiverse 15 2.3 Recap . 16 3 Approaches to New Axioms 17 3.1 Large Cardinals . 17 3.2 Inner Model Theory . 25 3.2.1 Basic Facts . 26 3.2.2 The Constructible Universe . 30 3.2.3 Other Inner Models . 35 3.2.4 Relative Constructibility . 38 3.3 Recap . 39 4 Ultimate L 40 4.1 The Axiom V = Ultimate L ..................... 41 4.2 Central Features of Ultimate L .................... 42 4.3 Further Philosophical Considerations . 47 4.4 Recap . 51 1 5 Set-theoretic Geology 52 5.1 Preliminaries . 52 5.2 The Downward Directed Grounds Hypothesis . 54 5.2.1 Bukovský’s Theorem . 54 5.2.2 The Main Argument . 61 5.3 Main Results . 65 5.4 Recap . 74 6 Conclusion 74 7 Appendix 75 7.1 Notation . 75 7.2 The ZFC Axioms . 76 7.3 The Ordinals . 77 7.4 The Universe of Sets . 77 7.5 Transitive Models and Absoluteness .
    [Show full text]
  • The Pure Symmetric Automorphisms of a Free Group Form a Duality Group
    THE PURE SYMMETRIC AUTOMORPHISMS OF A FREE GROUP FORM A DUALITY GROUP NOEL BRADY, JON MCCAMMOND, JOHN MEIER, AND ANDY MILLER Abstract. The pure symmetric automorphism group of a finitely generated free group consists of those automorphisms which send each standard generator to a conjugate of itself. We prove that these groups are duality groups. 1. Introduction Let Fn be a finite rank free group with fixed free basis X = fx1; : : : ; xng. The symmetric automorphism group of Fn, hereafter denoted Σn, consists of those au- tomorphisms that send each xi 2 X to a conjugate of some xj 2 X. The pure symmetric automorphism group, denoted PΣn, is the index n! subgroup of Σn of symmetric automorphisms that send each xi 2 X to a conjugate of itself. The quotient of PΣn by the inner automorphisms of Fn will be denoted OPΣn. In this note we prove: Theorem 1.1. The group OPΣn is a duality group of dimension n − 2. Corollary 1.2. The group PΣn is a duality group of dimension n − 1, hence Σn is a virtual duality group of dimension n − 1. (In fact we establish slightly more: the dualizing module in both cases is -free.) Corollary 1.2 follows immediately from Theorem 1.1 since Fn is a 1-dimensional duality group, there is a short exact sequence 1 ! Fn ! PΣn ! OPΣn ! 1 and any duality-by-duality group is a duality group whose dimension is the sum of the dimensions of its constituents (see Theorem 9.10 in [2]). That the virtual cohomological dimension of Σn is n − 1 was previously estab- lished by Collins in [9].
    [Show full text]
  • Direct Products and Homomorphisms
    Commuting properties Direct products and homomorphisms Simion Breaz logo Simion Breaz Direct products and homomorphisms Products and coproducts Commuting properties Contravariant functors Covariant functors Outline 1 Commuting properties Products and coproducts Contravariant functors Covariant functors logo Simion Breaz Direct products and homomorphisms Products and coproducts Commuting properties Contravariant functors Covariant functors Introduction Important properties of objects in particular categories (e.g. varieties of universal algebras) can be described using commuting properties of some canonical functors. For instance, in [Ad´amek and Rosicki: Locally presentable categories] there are the following examples: If V is a variety of finitary algebras and A ∈ V then A is finitely generated iff the functor Hom(A, −): V → Set preserves direct unions (i.e. directed colimits of monomorphisms); A is finitely presented (i.e. it is generated by finitely many generators modulo finitely many relations) iff the functor Hom(A, −): V → Set preserves directed colimits. logo Simion Breaz Direct products and homomorphisms Products and coproducts Commuting properties Contravariant functors Covariant functors Introduction Important properties of objects in particular categories (e.g. varieties of universal algebras) can be described using commuting properties of some canonical functors. For instance, in [Ad´amek and Rosicki: Locally presentable categories] there are the following examples: If V is a variety of finitary algebras and A ∈ V then A is finitely generated iff the functor Hom(A, −): V → Set preserves direct unions (i.e. directed colimits of monomorphisms); A is finitely presented (i.e. it is generated by finitely many generators modulo finitely many relations) iff the functor Hom(A, −): V → Set preserves directed colimits.
    [Show full text]
  • Generalized Quaternions
    GENERALIZED QUATERNIONS KEITH CONRAD 1. introduction The quaternion group Q8 is one of the two non-abelian groups of size 8 (up to isomor- phism). The other one, D4, can be constructed as a semi-direct product: ∼ ∼ × ∼ D4 = Aff(Z=(4)) = Z=(4) o (Z=(4)) = Z=(4) o Z=(2); where the elements of Z=(2) act on Z=(4) as the identity and negation. While Q8 is not a semi-direct product, it can be constructed as the quotient group of a semi-direct product. We will see how this is done in Section2 and then jazz up the construction in Section3 to make an infinite family of similar groups with Q8 as the simplest member. In Section4 we will compare this family with the dihedral groups and see how it fits into a bigger picture. 2. The quaternion group from a semi-direct product The group Q8 is built out of its subgroups hii and hji with the overlapping condition i2 = j2 = −1 and the conjugacy relation jij−1 = −i = i−1. More generally, for odd a we have jaij−a = −i = i−1, while for even a we have jaij−a = i. We can combine these into the single formula a (2.1) jaij−a = i(−1) for all a 2 Z. These relations suggest the following way to construct the group Q8. Theorem 2.1. Let H = Z=(4) o Z=(4), where (a; b)(c; d) = (a + (−1)bc; b + d); ∼ The element (2; 2) in H has order 2, lies in the center, and H=h(2; 2)i = Q8.
    [Show full text]
  • Some First-Order Probability Logics
    Theoretical Computer Science 247 (2000) 191–212 www.elsevier.com/locate/tcs View metadata, citation and similar papers at core.ac.uk brought to you by CORE Some ÿrst-order probability logics provided by Elsevier - Publisher Connector Zoran Ognjanovic a;∗, Miodrag RaÄskovic b a MatematickiÄ institut, Kneza Mihaila 35, 11000 Beograd, Yugoslavia b Prirodno-matematickiÄ fakultet, R. Domanovicaà 12, 34000 Kragujevac, Yugoslavia Received June 1998 Communicated by M. Nivat Abstract We present some ÿrst-order probability logics. The logics allow making statements such as P¿s , with the intended meaning “the probability of truthfulness of is greater than or equal to s”. We describe the corresponding probability models. We give a sound and complete inÿnitary axiomatic system for the most general of our logics, while for some restrictions of this logic we provide ÿnitary axiomatic systems. We study the decidability of our logics. We discuss some of the related papers. c 2000 Elsevier Science B.V. All rights reserved. Keywords: First order logic; Probability; Possible worlds; Completeness 1. Introduction In recent years there is a growing interest in uncertainty reasoning. A part of inves- tigation concerns its formal framework – probability logics [1, 2, 5, 6, 9, 13, 15, 17–20]. Probability languages are obtained by adding probability operators of the form (in our notation) P¿s to classical languages. The probability logics allow making formulas such as P¿s , with the intended meaning “the probability of truthfulness of is greater than or equal to s”. Probability models similar to Kripke models are used to give semantics to the probability formulas so that interpreted formulas are either true or false.
    [Show full text]
  • Lecture 1.3: Direct Products and Sums
    Lecture 1.3: Direct products and sums Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 8530, Advanced Linear Algebra M. Macauley (Clemson) Lecture 1.3: Direct products and sums Math 8530, Advanced Linear Algebra 1 / 5 Overview In previous lectures, we learned about vectors spaces and subspaces. We learned about what it meant for a subset to span, to be linearly independent, and to be a basis. In this lecture, we will see how to create new vector spaces from old ones. We will see several ways to \multiply" vector spaces together, and will learn how to construct: the complement of a subspace the direct sum of two subspaces the direct product of two vector spaces M. Macauley (Clemson) Lecture 1.3: Direct products and sums Math 8530, Advanced Linear Algebra 2 / 5 Complements and direct sums Theorem 1.5 (a) Every subspace Y of a finite-dimensional vector space X is finite-dimensional. (b) Every subspace Y has a complement in X : another subspace Z such that every vector x 2 X can be written uniquely as x = y + z; y 2 Y ; z 2 Z; dim X = dim Y + dim Z: Proof Definition X is the direct sum of subspaces Y and Z that are complements of each other. More generally, X is the direct sum of subspaces Y1;:::; Ym if every x 2 X can be expressed uniquely as x = y1 + ··· + ym; yi 2 Yi : We denote this as X = Y1 ⊕ · · · ⊕ Ym. M. Macauley (Clemson) Lecture 1.3: Direct products and sums Math 8530, Advanced Linear Algebra 3 / 5 Direct products Definition The direct product of X1 and X2 is the vector space X1 × X2 := (x1; x2) j x1 2 X1; x2 2 X2 ; with addition and multiplication defined component-wise.
    [Show full text]
  • 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].
    [Show full text]