DOCSLIB.ORG

Basic Concepts Page 1 of 39

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Basic Concepts Page 1 of 39 Hardegree, Set Theory, Chapter 1: Basic Concepts page 1 of 39 Basic 1 Concepts 1. Membership .....................................................................................................................................2 2. Pure Set Theory versus Impure Set Theory .....................................................................................2 3. The Axiom of Extensionality...........................................................................................................4 4. Inclusion and Exclusion...................................................................................................................5 5. Set-Abstracts....................................................................................................................................7 6. Whether Set-Abstracts Denote.........................................................................................................8 7. Classical Set Theory ......................................................................................................................11 8. Russell’s Paradox...........................................................................................................................12 9. Modern Set Theory........................................................................................................................13 10. The Axiom of Separation...............................................................................................................14 11. Russell’s Paradox Revisited...........................................................................................................15 12. The Empty Set................................................................................................................................16 13. Intersection and Set Difference......................................................................................................17 14. Union and Boolean Sum ................................................................................................................18 15. Absolute versus Relative Complementation..................................................................................19 16. Singletons, Doubletons, etc............................................................................................................20 17. General Union and Intersection .....................................................................................................21 18. Power Sets......................................................................................................................................23 19. Fields of Sets..................................................................................................................................24 20. The Principle of Set-Abstraction and Contextual Definitions .......................................................25 1. Principle of Set-Abstraction...............................................................................................25 2. The Move from Explicit Definitions to Implicit Definitions.............................................26 3. Contextual Definitions.......................................................................................................26 4. An Aside on Contextuality.................................................................................................27 21. Axioms for Chapter 1.....................................................................................................................28 22. Definitions for Chapter 1 ...............................................................................................................29 1. Official (Explicit) Definitions............................................................................................29 2. Grammatical Categories.....................................................................................................30 3. Contextual Definitions.......................................................................................................30 23. Theorems for Chapter 1 .................................................................................................................31 24. Exercises for Chapter 1..................................................................................................................33 1. Part 1:.................................................................................................................................33 2. Part 2:.................................................................................................................................33 3. Part 3:.................................................................................................................................33 4. Part 4:.................................................................................................................................33 5. Part 5..................................................................................................................................33 25. Answers to Exercises for Chapter 1...............................................................................................34 26. Examples of Derivations of Theorems ..........................................................................................35 Hardegree, Set Theory, Chapter 1: Basic Concepts page 2 of 39 1. Membership Basic to set theory is the notion of membership or elementhood. Indeed, set theory can be formulated in its entirety as a formal theory in which the only non-logical notion (symbol) is membership. The remaining notions can all be defined in terms of membership using the standard machinery of first-order logic (including identity and definite descriptions). Although set theory can be formulated exclusively in terms of membership, in actual practice, free use is made of defined expressions. Accordingly, very few set-theoretic theorems are actually displayed in primitive notation. Membership is a two-place relation, and its syntactic counterpart is a two-place predicate – (epsilon). Following the usual mathematical custom, we write this predicate in infix notation, rather than in prefix notation (as is customary in abstract logic and syntax). Specifically, in order to say that one thing is a member (element) of another thing, we write their respective names, and we infix the membership predicate, . Thus, the formula,1 may be read in either of the following ways. is a member of is an element of Infixing epsilon is analogous to the way the identity relation is expressed by the formula ‘’. Also, in analogy with the identity predicate, we simplify negated formulas in the natural way, according to the following definitions. (d1) [ ] (d2) [ ] Here, ‘’ and ‘’ stand for arbitrary singular terms. The expression ‘’ is a metalinguistic expression used to indicate that expressions of one form abbreviate expressions of another form. 2. Pure Set Theory versus Impure Set Theory In the previous section, in saying that membership is the only non-logical notion of set theory, I mean pure set theory. Pure set theory may be formulated as a formal theory in a first-order language in which the only non-logical symbol is ‘’. This is because pure set theory talks about sets, and nothing else. Thus, in particular, in the formula ‘ ’, both and are sets. By contrast, impure set theory (for lack of a better name!) talks not only about sets, but also about other things, including first elements and classes (not necessarily both). Briefly, the notion of class is a generalization of the notion of set; sets are special kinds of classes (see below). Anything that isn’t a class is a first element, sometimes called a point. First elements, or points, which are very important in informal set theory, have no internal structure from the viewpoint of set theory. Of course, 1 It depends upon the author whether ‘’ and ‘=’ are treated like other two-place functors written in infix format, specifically whether the official formula has outer parentheses. We adopt the view that parentheses (or brackets) officially flank identity statements and membership statements, and that these are dropped when convenient. Hardegree, Set Theory, Chapter 1: Basic Concepts page 3 of 39 first elements may have very interesting internal structure from the viewpoint of, say, biology! The point is that they have no internal set theoretic structure. In informal set theory, as used throughout modern mathematics, points include numbers, geometrical points, vectors, group elements, etc. Points also include all concrete objects, as well as all abstract objects (other than classes!) In one particular formulation of impure set theory, there are two non-logical predicates; besides ‘’ there is the one-place predicate ‘C’ (‘...is a class’). Three other predicates may then be defined in terms of ‘C’ and ‘’, as follows. F C S C & [ ] P C & [ ] [Here, is any singular-term, and is any variable not free in .] The predicate letters have the following intended readings. C : is a class; F : is a first element; S : is a set; P : is a proper class. A proper class is a class that is not a set, by the following theorem, which follows immediately from the definitions. (P [C & S]) Alternatively, we could introduce three primitive one-place predicates, ‘C’, ‘S’, and ‘P’, and add the following as axioms. (S [C & [ ]]); (P [C & [ ]]). Other than elegance, nothing hinges on whether we introduce additional terms by adding definitions or by adding axioms. Officially,
Load more

Recommended publications
  • Set Theory, by Thomas Jech, Academic Press, New York, 1978, Xii + 621 Pp., '$53.00
    BOOK REVIEWS 775 BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 3, Number 1, July 1980 © 1980 American Mathematical Society 0002-9904/80/0000-0 319/$01.75 Set theory, by Thomas Jech, Academic Press, New York, 1978, xii + 621 pp., '$53.00. "General set theory is pretty trivial stuff really" (Halmos; see [H, p. vi]). At least, with the hindsight afforded by Cantor, Zermelo, and others, it is pretty trivial to do the following. First, write down a list of axioms about sets and membership, enunciating some "obviously true" set-theoretic principles; the most popular Hst today is called ZFC (the Zermelo-Fraenkel axioms with the axiom of Choice). Next, explain how, from ZFC, one may derive all of conventional mathematics, including the general theory of transfinite cardi­ nals and ordinals. This "trivial" part of set theory is well covered in standard texts, such as [E] or [H]. Jech's book is an introduction to the "nontrivial" part. Now, nontrivial set theory may be roughly divided into two general areas. The first area, classical set theory, is a direct outgrowth of Cantor's work. Cantor set down the basic properties of cardinal numbers. In particular, he showed that if K is a cardinal number, then 2", or exp(/c), is a cardinal strictly larger than K (if A is a set of size K, 2* is the cardinality of the family of all subsets of A). Now starting with a cardinal K, we may form larger cardinals exp(ic), exp2(ic) = exp(exp(fc)), exp3(ic) = exp(exp2(ic)), and in fact this may be continued through the transfinite to form expa(»c) for every ordinal number a.
    [Show full text]
  • 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).
    [Show full text]
  • Automating Free Logic in HOL, with an Experimental Application in Category Theory
    Noname manuscript No. (will be inserted by the editor) Automating Free Logic in HOL, with an Experimental Application in Category Theory Christoph Benzm¨uller and Dana S. Scott Received: date / Accepted: date Abstract A shallow semantical embedding of free logic in classical higher- order logic is presented, which enables the off-the-shelf application of higher- order interactive and automated theorem provers for the formalisation and verification of free logic theories. Subsequently, this approach is applied to a selected domain of mathematics: starting from a generalization of the standard axioms for a monoid we present a stepwise development of various, mutually equivalent foundational axiom systems for category theory. As a side-effect of this work some (minor) issues in a prominent category theory textbook have been revealed. The purpose of this article is not to claim any novel results in category the- ory, but to demonstrate an elegant way to “implement” and utilize interactive and automated reasoning in free logic, and to present illustrative experiments. Keywords Free Logic · Classical Higher-Order Logic · Category Theory · Interactive and Automated Theorem Proving 1 Introduction Partiality and undefinedness are prominent challenges in various areas of math- ematics and computer science. Unfortunately, however, modern proof assistant systems and automated theorem provers based on traditional classical or intu- itionistic logics provide rather inadequate support for these challenge concepts. Free logic [24,25,30,32] offers a theoretically appealing solution, but it has been considered as rather unsuited towards practical utilization. Christoph Benzm¨uller Freie Universit¨at Berlin, Berlin, Germany & University of Luxembourg, Luxembourg E-mail: [email protected] Dana S.
    [Show full text]
  • Mathematical Logic. Introduction. by Vilnis Detlovs And
    1 Version released: May 24, 2017 Introduction to Mathematical Logic Hyper-textbook for students by Vilnis Detlovs, Dr. math., and Karlis Podnieks, Dr. math. University of Latvia This work is licensed under a Creative Commons License and is copyrighted © 2000-2017 by us, Vilnis Detlovs and Karlis Podnieks. Sections 1, 2, 3 of this book represent an extended translation of the corresponding chapters of the book: V. Detlovs, Elements of Mathematical Logic, Riga, University of Latvia, 1964, 252 pp. (in Latvian). With kind permission of Dr. Detlovs. Vilnis Detlovs. Memorial Page In preparation – forever (however, since 2000, used successfully in a real logic course for computer science students). This hyper-textbook contains links to: Wikipedia, the free encyclopedia; MacTutor History of Mathematics archive of the University of St Andrews; MathWorld of Wolfram Research. 2 Table of Contents References..........................................................................................................3 1. Introduction. What Is Logic, Really?.............................................................4 1.1. Total Formalization is Possible!..............................................................5 1.2. Predicate Languages.............................................................................10 1.3. Axioms of Logic: Minimal System, Constructive System and Classical System..........................................................................................................27 1.4. The Flavor of Proving Directly.............................................................40
    [Show full text]
  • The Logic of Brouwer and Heyting
    THE LOGIC OF BROUWER AND HEYTING Joan Rand Moschovakis Intuitionistic logic consists of the principles of reasoning which were used in- formally by L. E. J. Brouwer, formalized by A. Heyting (also partially by V. Glivenko), interpreted by A. Kolmogorov, and studied by G. Gentzen and K. G¨odel during the first third of the twentieth century. Formally, intuitionistic first- order predicate logic is a proper subsystem of classical logic, obtained by replacing the law of excluded middle A ∨¬A by ¬A ⊃ (A ⊃ B); it has infinitely many dis- tinct consistent axiomatic extensions, each necessarily contained in classical logic (which is axiomatically complete). However, intuitionistic second-order logic is inconsistent with classical second-order logic. In terms of expressibility, the intu- itionistic logic and language are richer than the classical; as G¨odel and Gentzen showed, classical first-order arithmetic can be faithfully translated into the nega- tive fragment of intuitionistic arithmetic, establishing proof-theoretical equivalence and clarifying the distinction between the classical and constructive consequences of mathematical axioms. The logic of Brouwer and Heyting is effective. The conclusion of an intuitionistic derivation holds with the same degree of constructivity as the premises. Any proof of a disjunction of two statements can be effectively transformed into a proof of one of the disjuncts, while any proof of an existential statement contains an effec- tive prescription for finding a witness. The negation of a statement is interpreted as asserting that the statement is not merely false but absurd, i.e., leads to a contradiction. Brouwer objected to the general law of excluded middle as claim- ing a priori that every mathematical problem has a solution, and to the general law ¬¬A ⊃ A of double negation as asserting that every consistent mathematical statement holds.
    [Show full text]
  • The Modal Logic of Potential Infinity, with an Application to Free Choice
    The Modal Logic of Potential Infinity, With an Application to Free Choice Sequences Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Ethan Brauer, B.A. ∼6 6 Graduate Program in Philosophy The Ohio State University 2020 Dissertation Committee: Professor Stewart Shapiro, Co-adviser Professor Neil Tennant, Co-adviser Professor Chris Miller Professor Chris Pincock c Ethan Brauer, 2020 Abstract This dissertation is a study of potential infinity in mathematics and its contrast with actual infinity. Roughly, an actual infinity is a completed infinite totality. By contrast, a collection is potentially infinite when it is possible to expand it beyond any finite limit, despite not being a completed, actual infinite totality. The concept of potential infinity thus involves a notion of possibility. On this basis, recent progress has been made in giving an account of potential infinity using the resources of modal logic. Part I of this dissertation studies what the right modal logic is for reasoning about potential infinity. I begin Part I by rehearsing an argument|which is due to Linnebo and which I partially endorse|that the right modal logic is S4.2. Under this assumption, Linnebo has shown that a natural translation of non-modal first-order logic into modal first- order logic is sound and faithful. I argue that for the philosophical purposes at stake, the modal logic in question should be free and extend Linnebo's result to this setting. I then identify a limitation to the argument for S4.2 being the right modal logic for potential infinity.
    [Show full text]
  • Dynamic Algebras: Examples, Constructions, Applications
    Dynamic Algebras: Examples, Constructions, Applications Vaughan Pratt∗ Dept. of Computer Science, Stanford, CA 94305 Abstract Dynamic algebras combine the classes of Boolean (B ∨ 0 0) and regu- lar (R ∪ ; ∗) algebras into a single finitely axiomatized variety (BR 3) resembling an R-module with “scalar” multiplication 3. The basic result is that ∗ is reflexive transitive closure, contrary to the intuition that this concept should require quantifiers for its definition. Using this result we give several examples of dynamic algebras arising naturally in connection with additive functions, binary relations, state trajectories, languages, and flowcharts. The main result is that free dynamic algebras are residu- ally finite (i.e. factor as a subdirect product of finite dynamic algebras), important because finite separable dynamic algebras are isomorphic to Kripke structures. Applications include a new completeness proof for the Segerberg axiomatization of propositional dynamic logic, and yet another notion of regular algebra. Key words: Dynamic algebra, logic, program verification, regular algebra. This paper originally appeared as Technical Memo #138, Laboratory for Computer Science, MIT, July 1979, and was subsequently revised, shortened, retitled, and published as [Pra80b]. After more than a decade of armtwisting I. N´emeti and H. Andr´eka finally persuaded the author to publish TM#138 itself on the ground that it contained a substantial body of interesting material that for the sake of brevity had been deleted from the STOC-80 version. The principal changes here to TM#138 are the addition of footnotes and the sections at the end on retrospective citations and reflections. ∗This research was supported by the National Science Foundation under NSF grant no.
    [Show full text]
  • On Families of Mutually Exclusive Sets
    ANNALS OF MATHEMATICS Vol. 44, No . 2, April, 1943 ON FAMILIES OF MUTUALLY EXCLUSIVE SETS BY P . ERDÖS AND A. TARSKI (Received August 11, 1942) In this paper we shall be concerned with a certain particular problem from the general theory of sets, namely with the problem of the existence of families of mutually exclusive sets with a maximal power . It will turn out-in a rather unexpected way that the solution of these problems essentially involves the notion of the so-called "inaccessible numbers ." In this connection we shall make some general remarks regarding inaccessible numbers in the last section of our paper . §1. FORMULATION OF THE PROBLEM . TERMINOLOGY' The problem in which we are interested can be stated as follows : Is it true that every field F of sets contains a family of mutually exclusive sets with a maximal power, i .e . a family O whose cardinal number is not smaller than the cardinal number of any other family of mutually exclusive sets contained in F . By a field of sets we understand here as usual a family F of sets which to- gether with every two sets X and Y contains also their union X U Y and their difference X - Y (i.e. the set of those elements of X which do not belong to Y) among its elements . A family O is called a family of mutually exclusive sets if no set X of X of O is empty and if any two different sets of O have an empty inter- section. A similar problem can be formulated for other families e .g .
    [Show full text]
  • Completeness
    Completeness The strange case of Dr. Skolem and Mr. G¨odel∗ Gabriele Lolli The completeness theorem has a history; such is the destiny of the impor- tant theorems, those for which for a long time one does not know (whether there is anything to prove and) what to prove. In its history, one can di- stinguish at least two main paths; the first one covers the slow and difficult comprehension of the problem in (what historians consider) the traditional development of mathematical logic canon, up to G¨odel'sproof in 1930; the second path follows the L¨owenheim-Skolem theorem. Although at certain points the two paths crossed each other, they started and continued with their own aims and problems. A classical topos of the history of mathema- tical logic concerns the how and the why L¨owenheim, Skolem and Herbrand did not discover the completeness theorem, though they proved it, or whe- ther they really proved, or perhaps they actually discovered, completeness. In following these two paths, we will not always respect strict chronology, keeping the two stories quite separate, until the crossing becomes decisive. In modern pre-mathematical logic, the notion of completeness does not appear. There are some interesting speculations in Kant which, by some stretching, could be realized as bearing some relation with the problem; Kant's remarks, however, are probably more related with incompleteness, in connection with his thoughts on the derivability of transcendental ideas (or concepts of reason) from categories (the intellect's concepts) through a pas- sage to the limit; thus, for instance, the causa prima, or the idea of causality, is the limit of implication, or God is the limit of disjunction, viz., the catego- ry of \comunance".
    [Show full text]
  • Sprinkles of Extensionality for Your Vanilla Type Theory
    Sprinkles of extensionality for your vanilla type theory Jesper Cockx1 and Andreas Abel2 1 DistriNet { KU Leuven 2 Department of Computer Science and Engineering { Gothenburg University Dependent types can make your developments (be they programs or proofs) dramatically safer by allowing you to prove that you implemented what you intended. Unfortunately, they can also make your developments dramatically more boring by requiring you to elaborate those proofs in often painstaking detail. For this reason, dependently typed languages typically allow you to cheat by postulating some facts as axioms. However, type theory is not just about which types are inhabited; it's about how things compute. Computationally, an axiom is a stuck term, so it prevents you from evaluating your programs. What if you could postulate not just axioms, but also arbitrary evaluation rules? 1. A typical frustration for people new to proof assistants like Agda or Coq is that 0 + x evaluates to x for arbitrary x, but x + 0 doesn't. Of course, a lemma for x + 0 = x is easy to prove, but having to appeal to this lemma explicitly in all subsequent uses is bothersome. By adding an evaluation rule x + 0 −! x, you can get the automatic application of this lemma. Similarly, you can add a rule x + suc y −! suc (x + y), or (x + y) + z −! x + (y + z). 2. Allais, McBride, and Boutillier[2013] present a series of evaluation rules of this type (which they call ν-rules) for functions on pairs and lists. For example, they have a rule for concatenating with an empty list (l ++ [] −! l), but also rules for simplifying expressions involving map and fold (e.g.
    [Show full text]
  • Equivalents to the Axiom of Choice and Their Uses A
    EQUIVALENTS TO THE AXIOM OF CHOICE AND THEIR USES A Thesis Presented to The Faculty of the Department of Mathematics California State University, Los Angeles In Partial Fulfillment of the Requirements for the Degree Master of Science in Mathematics By James Szufu Yang c 2015 James Szufu Yang ALL RIGHTS RESERVED ii The thesis of James Szufu Yang is approved. Mike Krebs, Ph.D. Kristin Webster, Ph.D. Michael Hoffman, Ph.D., Committee Chair Grant Fraser, Ph.D., Department Chair California State University, Los Angeles June 2015 iii ABSTRACT Equivalents to the Axiom of Choice and Their Uses By James Szufu Yang In set theory, the Axiom of Choice (AC) was formulated in 1904 by Ernst Zermelo. It is an addition to the older Zermelo-Fraenkel (ZF) set theory. We call it Zermelo-Fraenkel set theory with the Axiom of Choice and abbreviate it as ZFC. This paper starts with an introduction to the foundations of ZFC set the- ory, which includes the Zermelo-Fraenkel axioms, partially ordered sets (posets), the Cartesian product, the Axiom of Choice, and their related proofs. It then intro- duces several equivalent forms of the Axiom of Choice and proves that they are all equivalent. In the end, equivalents to the Axiom of Choice are used to prove a few fundamental theorems in set theory, linear analysis, and abstract algebra. This paper is concluded by a brief review of the work in it, followed by a few points of interest for further study in mathematics and/or set theory. iv ACKNOWLEDGMENTS Between the two department requirements to complete a master's degree in mathematics − the comprehensive exams and a thesis, I really wanted to experience doing a research and writing a serious academic paper.
    [Show full text]
  • The Structure and Ontology of Logical Space June 1983
    WORLDS AND PROPOSITIONS: The Structure and Ontology of Logical Space Phillip Bricker A DISSERTATION PRESENTED TO THE FACULTY OF PRINCETON UNIVERSITY IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PH I LOSOPHY RECOMMENDED FOR ACCEPTANCE BY THE DEPARTMENT OF PHILOSOPHY June 1983 © 1983 Phillip Bricker ALL RIGHTS RESERVED ACKNOWLEDGMENTS I acknowledge with pleasure an enormous intellectual debt to my thesis adviser, David Lewis, both for his extensive and invaluable comments on various drafts of this work, and, more generally, for teaching me so much of what I know about matters philosophical. CONTENTS Introduction. 3 Section 1: Propositions and Boolean Algebra. 10 First Thesis. 10 First Reformulation: The Laws of Propositional Logic. 15 Second Reformulation: The Lindenbaum-Tarski Algebra. 21 Objections to the First Thesis. 28 Second Thesis. 33 Objections to the Second Thesis. 3& section 2: Logical Space. 38 Logical Space as a Field of Sets. 38 Alternative Conceptions of Logical Space. 44 section 3: Maximal Consistent Sets of Propositions. 48 Third Thesis. 48 Atomic Propositions. 52 Finite Consistency. 55 Objections to the Third Thesis. 59 section 4: Separating Worlds by Propositions. 68 Fourth Thesis. 68 Indiscernibility Principles. 71 Objections to the Fourth Thesis. 74 Section 5: Consistency and Realizability. 78 Fifth Thesis. 78 The Compact Theory. 82 The Mathematical Objection. 91 The Modal Objection. 105 Section 6: Proposition-Based Theories. 117 Tripartite Correspondence. 117 Weak and Feeble Proposition-Based Theories. 123 Adams's World-Story Theory. 131 Section 7: The World-Based Theory. 138 Isomorphic Algebras. 138 Stalnaker on Adams. 148 Section 8: Reducing possible Worlds to Language. 156 Introduction.
    [Show full text]