DOCSLIB.ORG

The Versatility of Universality in Principia Mathematica

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Versatility of Universality in Principia Mathematica January 2, 2011 11:18 History and Philosophy of Logic types HISTORY AND PHILOSOPHY OF LOGIC, 00 (Month 200x), 1–21 The Versatility of Universality in Principia Mathematica Brice Halimi Paris Ouest University, France Received 00 Month 200x; final version received 00 Month 200x In this article, I examine the ramified type theory set out in the first edition of Russell and Whitehead’s Principia Mathematica. My starting point is the “no loss of generality” problem: Russell, in the Introduction ((Russell and Whitehead 1910), p. 53-54), says that one can account for all propositional functions using predicative variables only, that is, dismissing non-predicative variables. That claim is not self-evident at all, hence a problem. The purpose of this article is to clarify Russell’s claim and to solve the “no loss of generality” problem. I first remark that the hierarchy of propositional functions calls for a fine-grained conception of ramified types as propositional forms (“ramif-types”). Then, comparing different important interpretations of Principia’s theory of types, I consider the question as to whether Principia allows for non-predicative propositional functions and variables thereof. I explain how the distinction between the formal system of the theory, on the one hand, and its realizations in different epistemic universes, on the other, makes it possible to give a more satisfactory answer to that question than those given by previous commentators, and, as a consequence, to solve the “no loss of generality” problem. The solution consists in a substitutional semantics for non-predicative variables and non-predicative complex terms, based on an epistemic understanding of the order component of ramified types. The rest of the article then develops that epistemic understanding, adding an original epistemic model theory to the formal system of types. This shows that the universality sought by Russell for logic does not preclude semantical considerations, contrary to what van Heijenoort and Hintikka have claimed. Keywords: Principia Mathematica, ramified type theory, predicativity, epistemic diagram, reducibility, logical universalism. 1. The Principia’s theory of types and the “no loss of generality” problem 1.1. The “no loss of generality” problem The Introduction of the (first edition of) Principia Mathematica suggests that in practice only pred- icative predicate variables are adopted in the formal language and that there is “no loss of generality” in doing so: It is important to observe that all possible functions in the above hierarchy [the hierarchy of functions of increasing orders] can be obtained by means of predicative functions and apparent variables. [. ] Thus we need not introduce as variables any functions except predicative functions1. This is what I shall refer to as the “dispensability argument,” according to which one can do with pred- icative variables only. Russell and Whitehead continue: It should be observed that, in virtue of the manner in which our hierarchy of functions was generated, non- predicative functions always result from such as are predicative by means of generalization. Hence it is unnecessary to introduce a special notation for non-predicative functions of a given order and taking ar- guments of a given order [. ]; the only functions which will be so used [as apparent variables] will be predicative functions, because, as we have just seen, this restriction involves no loss of generality2. This is odd and requires explanation. How can there be no loss of generality? If non-predicative variables range over non-predicative attributes, then there is certainly a loss of generality. The solution of that problem is the main purpose of this paper. Before analyzing and explaining in greater detail the notions at stake here (“order,” “predicate vari- able,” “predicative,” “non-predicative”), let me set out the principle of the solution of the problem which I advocate. Were Principia to adopt non-predicative terms (variables and complex predicates obtained 1(Russell and Whitehead 1910), p. 53-54. 2(Russell and Whitehead 1910), p. 165. History and Philosophy of Logic ISSN: 0144-5340 print/ISSN 1464-5149 online © 200x Taylor & Francis http://www.informaworld.com DOI: 10.1080/0144534YYxxxxxxxx January 2, 2011 11:18 History and Philosophy of Logic types 2 by circumflexion), it would adopt a substitutional semantics for them. This substitutional semantics treats such terms as dummy schematic letters for wffs of a certain fixed structure matching the order component of the predicate terms. By the lights of a theory without such terms, these terms might be regarded as non-predicative. This is a non-standard use of “non-predicative” which normally means that comprehension axioms of the form ( E '(t)) (xt) '(t)(xt) ≡ A, where '(t) is not free in A, are adopted which make no further distinctions between the wffs allowed in A. Comprehension axioms involving non-predicative variables such as ( E 'h2;oi) (xo) 'h2;(o)i(xo) ≡ A (where h2; oi indicates that ' is an individual-level predicate variable of order 2)3 require that the wff A match in syntactic structure to the order 2 (i.e., no bound predicate variables of order 2 or higher may occur in A). I will argue that this substitutional semantics has to be understood epistemically. For example, some epistemic subject, other than me, mentions (what are to her) predicative attributes with which, however, I am not acquainted: I am nonetheless able to interpret these in my epistemic universe, but, in doing so, I cannot but understand them as involving some quantification that makes them non-predicative. I will then use a non-predicative functional term F (say, of type t) only as a proxy for what is, in some other epistemic perspective than mine, a predicative attribute. Accordingly, any formula ('t) A(') containing an apparent non-predicative variable of type t is interpreted as asserting that every formula A[F='] is true for any non-predicative such term F . Now given the substitutional semantics for non-predicate variables and circumflex predicate terms, a predicative and a non-predicative term of the same simple type which are provably extensionally equiva- lent, are not semantically distinct. In effect, under the substitutional semantics, the order amounts to just the order of the simple type (which is Principia’s syntactic definition of predicativity). Thus we have a way of explaining Principia’s otherwise perplexing statement that there is no loss of generality by omit- ting non-predicative terms from the language. The difference, in the context of a substitutional semantics, is epistemic (not ontological), in the sense that I have just suggested. So, for example, one can introduce h1;oi F zb := (' ) χ('; zb). This is a non-predicative functional term of order 2. Accordingly, one can use a non-predicative variable 'h2;oi having F as a possible substituend. Now suppose A('h2;oi) is, for E instance, E h3;h2;oii (zo) 'z ⊃ ('). Then A(F ) is h3;h2;oii ('h1;oi) (zo) χ('; z) ⊃ (F ), where the apparent variable is predicative (since it is of order 3 and has predicates of order 2 as arguments). Sentences such as A(F ) are substitutional instances of the non-predicatively quantified ('t) A('), and represent the counterpart in my epistemic universe of the use of bound predicative vari- ables in some other epistemic universe when the presumed values of those variables are not accessible to me as predicative attributes. In order to make that clear, we should first explain how orders and types are introduced in Principia and to what extent they constitute different features (section 1), then examine three preliminary questions to set out the main interpretations of orders and types in Principia’s syntax, before putting forward a solution to the “no loss of generality” problem that both supposes a substitutional interpretation of non-predicative variables (section 2) and leads to a new epistemic perspective on Principia (section 3). 1.2. The Vicious Circle Principle and the derivation of types The Introduction of the (first edition of) Principia Mathematica gives roughly three versions of the Vicious Circle Principle (VCP). By itself, this principle points at the vicious circle arising as soon as a collection of objects is supposed to include some member which cannot be defined otherwise than by referring to this very collection. The addition of the clause (C1), according to which a propositional function presupposes the totality of its values, gives rise to a new form (VCP 1) of the VCP: any propo- sitional function cannot have values which involve it. Inasmuch as any argument of any propositional function is a constituent of the corresponding value of this function (C2), no propositional function can be involved in its arguments (VCP 2). Finally, since ‘(x) F x’ involves F x^ (C3), the proposition (x) F x 3Here the index is the concatenation of the order and of the simple type. Later on, I will correct and refine the index so as to obtain genuine ramified-type indices. January 2, 2011 11:18 History and Philosophy of Logic types 3 cannot be an argument for F (VCP 3). To sum up: VCP 3 = VCP 2 + C3 = VCP 1 + C2 + C3 = VCP + C1 + C2 + C3. This is this final version (VCP 3) to which I will refer when speaking of the “vicious circle principle.” After formulating these successive versions of the VCP4, which he sets out rather abstractly, Russell illustrates it with the case of the truth value of the proposition ‘(p) p is false’5. The Principles of Math- ematics would claim that this proposition must be false, and therefore have a truth value, simply because its truth is self-refuting. Of course, this solution is no more practicable6. Nevertheless, Russell seems to hold on to the basic essential: the proposition cannot be devoid of any truth value.
Load more

Recommended publications
  • Basic Concepts of Set Theory, Functions and Relations 1. Basic
    Ling 310, adapted from UMass Ling 409, Partee lecture notes March 1, 2006 p. 1 Basic Concepts of Set Theory, Functions and Relations 1. Basic Concepts of Set Theory........................................................................................................................1 1.1. Sets and elements ...................................................................................................................................1 1.2. Specification of sets ...............................................................................................................................2 1.3. Identity and cardinality ..........................................................................................................................3 1.4. Subsets ...................................................................................................................................................4 1.5. Power sets .............................................................................................................................................4 1.6. Operations on sets: union, intersection...................................................................................................4 1.7 More operations on sets: difference, complement...................................................................................5 1.8. Set-theoretic equalities ...........................................................................................................................5 Chapter 2. Relations and Functions ..................................................................................................................6
    [Show full text]
  • Are Large Cardinal Axioms Restrictive?
    Are Large Cardinal Axioms Restrictive? Neil Barton∗ 24 June 2020y Abstract The independence phenomenon in set theory, while perva- sive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles. In this paper, I argue that whether or not large cardinal axioms count as maximality prin- ciples depends on prior commitments concerning the richness of the subset forming operation. In particular I argue that there is a conception of maximality through absoluteness, on which large cardinal axioms are restrictive. I argue, however, that large cardi- nals are still important axioms of set theory and can play many of their usual foundational roles. Introduction Large cardinal axioms are widely viewed as some of the best candi- dates for new axioms of set theory. They are (apparently) linearly ordered by consistency strength, have substantial mathematical con- sequences for questions independent from ZFC (such as consistency statements and Projective Determinacy1), and appear natural to the ∗Fachbereich Philosophie, University of Konstanz. E-mail: neil.barton@uni- konstanz.de. yI would like to thank David Aspero,´ David Fernandez-Bret´ on,´ Monroe Eskew, Sy-David Friedman, Victoria Gitman, Luca Incurvati, Michael Potter, Chris Scam- bler, Giorgio Venturi, Matteo Viale, Kameryn Williams and audiences in Cambridge, New York, Konstanz, and Sao˜ Paulo for helpful discussion. Two anonymous ref- erees also provided helpful comments, and I am grateful for their input. I am also very grateful for the generous support of the FWF (Austrian Science Fund) through Project P 28420 (The Hyperuniverse Programme) and the VolkswagenStiftung through the project Forcing: Conceptual Change in the Foundations of Mathematics.
    [Show full text]
  • THE 1910 PRINCIPIA's THEORY of FUNCTIONS and CLASSES and the THEORY of DESCRIPTIONS*
    EUJAP VOL. 3 No. 2 2007 ORIGinal SCienTifiC papeR UDK: 165 THE 1910 PRINCIPIA’S THEORY OF FUNCTIONS AND CLASSES AND THE THEORY OF DESCRIPTIONS* WILLIAM DEMOPOULOS** The University of Western Ontario ABSTRACT 1. Introduction It is generally acknowledged that the 1910 Prin- The 19101 Principia’s theory of proposi- cipia does not deny the existence of classes, but tional functions and classes is officially claims only that the theory it advances can be developed so that any apparent commitment to a “no-classes theory of classes,” a theory them is eliminable by the method of contextual according to which classes are eliminable. analysis. The application of contextual analysis But it is clear from Principia’s solution to ontological questions is widely viewed as the to the class paradoxes that although the central philosophical innovation of Russell’s theory of descriptions. Principia’s “no-classes theory it advances holds that classes are theory of classes” is a striking example of such eliminable, it does not deny their exis- an application. The present paper develops a re- tence. Whitehead and Russell argue from construction of Principia’s theory of functions the supposition that classes involve or and classes that is based on Russell’s epistemo- logical applications of the method of contextual presuppose propositional functions to the analysis. Such a reconstruction is not eliminativ- conclusion that the paradoxical classes ist—indeed, it explicitly assumes the existence of are excluded by the nature of such func- classes—and possesses certain advantages over tions. This supposition rests on the repre- the no–classes theory advocated by Whitehead and Russell.
    [Show full text]
  • How Peircean Was the “'Fregean' Revolution” in Logic?
    HOW PEIRCEAN WAS THE “‘FREGEAN’ REVOLUTION” IN LOGIC? Irving H. Anellis Peirce Edition, Institute for American Thought Indiana University – Purdue University at Indianapolis Indianapolis, IN, USA [email protected] Abstract. The historiography of logic conceives of a Fregean revolution in which modern mathematical logic (also called symbolic logic) has replaced Aristotelian logic. The preeminent expositors of this conception are Jean van Heijenoort (1912–1986) and Don- ald Angus Gillies. The innovations and characteristics that comprise mathematical logic and distinguish it from Aristotelian logic, according to this conception, created ex nihlo by Gottlob Frege (1848–1925) in his Begriffsschrift of 1879, and with Bertrand Rus- sell (1872–1970) as its chief This position likewise understands the algebraic logic of Augustus De Morgan (1806–1871), George Boole (1815–1864), Charles Sanders Peirce (1838–1914), and Ernst Schröder (1841–1902) as belonging to the Aristotelian tradi- tion. The “Booleans” are understood, from this vantage point, to merely have rewritten Aristotelian syllogistic in algebraic guise. The most detailed listing and elaboration of Frege’s innovations, and the characteristics that distinguish mathematical logic from Aristotelian logic, were set forth by van Heijenoort. I consider each of the elements of van Heijenoort’s list and note the extent to which Peirce had also developed each of these aspects of logic. I also consider the extent to which Peirce and Frege were aware of, and may have influenced, one another’s logical writings. AMS (MOS) 2010 subject classifications: Primary: 03-03, 03A05, 03C05, 03C10, 03G27, 01A55; secondary: 03B05, 03B10, 03E30, 08A20; Key words and phrases: Peirce, abstract algebraic logic; propositional logic; first-order logic; quantifier elimina- tion, equational classes, relational systems §0.
    [Show full text]
  • Foundations of Mathematics
    Foundations of Mathematics. Does mathematics need foundations? (Not until 1900.) Mathematical approach: Work towards an axiom system of mathematics with purely mathematical means. ( Hilbert’s Programme ). In its naïve interpretation crushed by Gödel’s Incompleteness Theorem. Extra-mathematical approach: Use external arguments for axioms and rules: pragmatic, philosophical, sociological, (theological ?). Foundations of number theory: test case. Core Logic – 2007/08-1ab – p. 2/36 Sets are everything (1). Different areas of mathematics use different primitive notions: ordered pair, function, natural number, real number, transformation, etc. Set theory is able to incorporate all of these in one framework: Ordered Pair. We define hx, y i := {{ x}, {x, y }} . (Kuratowski pair ) Function. A set f is called a function if there are sets X and Y such that f ⊆ X × Y and ′ ′ ′ ∀x, y, y hx, y i ∈ f&hx, y i ∈ f → y = y . Core Logic – 2007/08-1ab – p. 3/36 Sets are everything (2). Set theory incorporates basic notions of mathematics: Natural Numbers. We call a set X inductive if it contains ∅ and for each x ∈ X, we have x ∪ { x} ∈ X. Assume that there is an inductive set. Then define N to be the intersection of all inductive sets. Rational Numbers. We define P := {0, 1} × N × N\{ 0}, then hi, n, m i ∼ h j, k, ℓ i : ⇐⇒ i = j & n · ℓ = m · k, and Q := P/∼. Core Logic – 2007/08-1ab – p. 4/36 Sets are everything (3). Set theory incorporates basic notions of mathematics: Real Numbers. Define an order on Q by hi, n, m i ≤ h j, k, ℓ i : ⇐⇒ i < j ∨ (i = j & n · ℓ ≤ k · m).
    [Show full text]
  • Russell's Paradox: Let a Be the Set of All Sets Which Do Not Contain
    Russell’s paradox: Let A be the set of all sets which do not contain themselves = {S | S 6∈ S} Ex: {1} ∈ {{1}, {1, 2}}, but {1} 6∈ {1} Is A∈A? Suppose A∈A. Then by definition of A, A 6∈ A. Suppose A 6∈ A. Then by definition of A, A∈A. Thus we need axioms in order to create mathematical objects. Principia Mathematica by Alfred North Whitehead and Bertrand Russell From: http://plato.stanford.edu/entries/principia-mathematica/ Logicism is the view that (some or all of) mathematics can be reduced to (formal) logic. It is often explained as a two-part thesis. First, it consists of the claim that all mathematical truths can be translated into logical truths or, in other words, that the vocabulary of mathematics constitutes a proper subset of the vocabulary of logic. Second, it consists of the claim that all mathematical proofs can be recast as logical proofs or, in other words, that the theorems of mathematics constitute a proper subset of the theorems of logic. In Bertrand Russell’s words, it is the logicist’s goal ”to show that all pure mathematics follows from purely logical premises and uses only concepts definable in logical terms.”[1] From: http://www.math.vanderbilt.edu/∼schectex/ccc/choice.html Axiom of Choice. Let C be a collection of nonempty sets. Then we can choose a member from each set in that collection. In other words, there exists a function f defined on C with the property that, for each set S in the collection, f(S) is a member of S.
    [Show full text]
  • The Substitutional Paradox in Russell's 1907 Letter to Hawtrey
    THE SUBSTITUTIONAL PARADOX IN RUSSELL’S 1907 LETTER TO HAWTREY B L Philosophy / U. of Alberta Edmonton, , Canada .@. This note presents a transcription of Russell’s letter to Hawtrey of January accompanied by some proposed emendations. In that letter Russell describes the paradox that he says “pilled” the “substitutional theory” devel- oped just before he turned to the theory of types. A close paraphrase of the deri- vation of the paradox in a contemporary Lemmon-style natural deduction system shows which axioms the theory must assume to govern its characteristic notion of substituting individuals and propositions for each other in other propositions. Other discussions of this paradox in the literature are mentioned. I conclude with remarks about the significance of the paradox for Russell. n the years to Bertrand Russell worked on what is now called the “substitutional theory” with its primitive notion of substi- Ituting one entity for another in a proposition, as a foundation for logic and source of a solution to the paradoxes. Russell abandoned that approach quite abruptly and returned to a logic based on propositional functions, eventually to appear as Principia Mathematica. Almost all of the material on the substitutional theory has remained unpublished; however, much will appear in print as the subject matter of Volume of the Collected Papers of Bertrand Russell. In recent years there has been some discussion of the substitutional theory, most prominently by Including, one may hope, a transcription of the letter which is the topic of this note. russell: the Journal of Bertrand Russell Studies n.s. (winter –): – The Bertrand Russell Research Centre, McMaster U.
    [Show full text]
  • Weyl's `Das Kontinuum' — 100 Years Later
    Arnon Avron Weyl's `Das Kontinuum' | 100 years later Orevkov'80 Conference St. Petersburg Days of Logic and Computability V April 2020 Prologue All platonists are alike; each anti-platonist is unhappy in her/his own way... My aim in this talk is first of all to present Weyl's views and system, at the time he wrote \Das Kontinuum" (exactly 100 years ago). Then I'll try to describe mine, which I believe are rather close to Weyl's original ideas (but still different). Weyl's Goals \I shall show that the house of analysis is to a large degree built on sand. I believe that I can replace this shifting foundation with pillars of enduring strength. They will not, however, support everything which today is generally considered to be securely grounded. I give up the rest, since I see no other possibility." \I would like to be understood . by all students who have become acquainted with the currently canonical and al- legedly `rigorous' foundations of analysis." \In spite of Dedekind, Cantor, and Weierstrass, the great task which has been facing us since the Pythagorean dis- covery of the irrationals remains today as unfinished as ever" Weyl and P´olya's Wager in 1918 Within 20 years P´olya and the majority of representative mathematicians will admit that the statements 1 Every bounded set of reals has a precise supremum 2 Every infinite set of numbers contains a denumerable subset contain totally vague concepts, such as \number," \set," and \denumerable," and therefore that their truth or fal- sity has the same status as that of the main propositions of Hegel's natural philosophy.
    [Show full text]
  • On Formally Undecidable Propositions of Principia Mathematica and Related Systems KURT GÖDEL
    On Formally Undecidable Propositions of Principia Mathematica And Related Systems KURT GÖDEL Translated by B. MELTZER Introduction by R. B. BRAITHWAITE TO CHRISTOPHER FERNAU in gratitude FL: Page 1 11/10/00 PREFACE Kurt Gödel’s astonishing discovery and proof, published in 1931, that even in elementary parts of arithmetic there exist propositions which cannot be proved or disproved within the system, is one of the most important contributions to logic since Aristotle. Any formal logical system which disposes of sufficient means to compass the addition and multiplication of positive integers and zero is subject to this limitation, so that one must consider this kind of incompleteness an inherent characteristic of formal mathematics as a whole, which was before this customarily considered the unequivocal intellectual discipline par excellence. No English translation of Gödel’s paper, which occupied twenty-five pages of the Monatshefte für Mathematik und Physik, has been generally available, and even the original German text is not everywhere easily accessible. The argument, which used a notation adapted from that of Whitehead and Russell’s Principia Mathematica, is a closely reasoned one and the present translation—besides being a long overdue act of piety—should make it more easily intelligible and much more widely read. In the former respect the reader will be greatly aided by the Introduction contributed by the Knightbridge Professor of Moral Philosophy in the University of Cambridge; for this is an excellent work of scholarship in its own right, not only pointing out the significance of Gödel’s work, but illuminating it by a paraphrase of the major part of the whole great argument.
    [Show full text]
  • The Propositional Logic of Principia Mathematica and Some of Its
    The propositional logic ofPM 93 2. Before and after Principia Mathematica One source which would have answered all questions is, of course, Russell. I imagine that he worked out all the demonstrations in detail and then condensed them to the current abbreviated form. At the least he must have had a sketch for The propositional logic of the complete demonstration. Presumably the next step would be to send them to Whitehead for comments. For a discussion of this process see Douglas Lackey's "The Whitehead Correspondence" [10]. In any case neither Russell's working Principia Mathematica and papers nor the letters to and responses from Whitehead concerning Part I, Section A are in the Russell Archives. There is however a letter from Whitehead to Russell some of its forerunners which indicates that, in an early draft of Part I, Section A, the demonstrations had not been written out in sufficient detail. Whitehead says, "Everything, even the object of the book, has been sacrificed to making the proofs look short and neat. It by Daniel J. O'Leary is essential, especially in the early parts that the proofs be written out fully-" (emphasis in original). This letter [31] had been later dated by Russell and a types­ cript [32]"prepared. On the bottom of the letter Russell has written, "Whitehead's criticism of *1-*5 ofPrincipia Mathematica". On the bottom of the typescript Rus­ sell wrote, "A criticism of my first draft of the Logic of Propositions for the begin­ ning of Principia Mathematica. Whitehead was entirely right." Russell assigned a I.
    [Show full text]
  • Kurt Gödel 1931
    Kurt Godel,¨ ‘Uber¨ formal unentscheidbare Satze¨ der Principia mathematica und verwandter Systeme I’ (1931) Richard Zach First publication: Monatshefte fur¨ Mathematik und Physik, 37, 173–198 Reprints: S. Feferman et al., eds., Kurt Godel.¨ Collected Works. Volume I: Publi- cations 1929–1936. New York: Oxford University Press, 1986, pp. 116–195. Translations: English translations: ‘On formally undecidable propositions of Prin- cipia mathematica and related systems I.’ Translation by B. Meltzer, On Formally Undecidable Propositions of Principia Mathematica and Related Systems, Edinburgh: Oliver and Boyd, 1962. Translation by E. Mendelsohn in M. Davis, The Undecidable, Hewlett, N.Y.: Raven Press, 1965, pp. 4–38. Translation and introduction by J. van Heijenoort in J. van Heijenoort (ed.), From Frege to Godel.¨ A Source Book in Mathe- matical Logic, Cambridge, Mass.: Harvard University Press, 1967, pp. 592–617. Van Heijenoort’s translation was approved by Godel¨ and is reprinted with an introduction by S. C. Kleene in S. Feferman et al., eds., Kurt Godel.¨ Collected Works. Volume I: Publications 1929–1936. New York: Oxford University Press, 1986, pp. 126– 195 (also contains several related pieces). Also reprinted in S. G. Shanker, Godel’s¨ Theorem in Focus, London: Routledge, 1988, pp. 17–47. Spanish translations: ‘So- bre proposiciones formalmente indecidibles de los Principia Mathematica y sistemas afines.’ Cuadernos Teorema 8. Valencia (Spain): Revista Teorema, 1980, by M. Gar- rido, A. Garc´ıa Suarez, and L. Ml. Valdes.;´ ‘Sobre sentencias formalmente inde- cidibles de Principa Matematica y sistemas afines,’ In: K. Godel,¨ Obras Completas, J. Moster´ın (ed.), Madrid: Alianza Editorial,1981, pp. 45–90.
    [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]