Chapter 1 Logic and Set Theory
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1 Elementary Set Theory
1 Elementary Set Theory Notation: fg enclose a set. f1; 2; 3g = f3; 2; 2; 1; 3g because a set is not defined by order or multiplicity. f0; 2; 4;:::g = fxjx is an even natural numberg because two ways of writing a set are equivalent. ; is the empty set. x 2 A denotes x is an element of A. N = f0; 1; 2;:::g are the natural numbers. Z = f:::; −2; −1; 0; 1; 2;:::g are the integers. m Q = f n jm; n 2 Z and n 6= 0g are the rational numbers. R are the real numbers. Axiom 1.1. Axiom of Extensionality Let A; B be sets. If (8x)x 2 A iff x 2 B then A = B. Definition 1.1 (Subset). Let A; B be sets. Then A is a subset of B, written A ⊆ B iff (8x) if x 2 A then x 2 B. Theorem 1.1. If A ⊆ B and B ⊆ A then A = B. Proof. Let x be arbitrary. Because A ⊆ B if x 2 A then x 2 B Because B ⊆ A if x 2 B then x 2 A Hence, x 2 A iff x 2 B, thus A = B. Definition 1.2 (Union). Let A; B be sets. The Union A [ B of A and B is defined by x 2 A [ B if x 2 A or x 2 B. Theorem 1.2. A [ (B [ C) = (A [ B) [ C Proof. Let x be arbitrary. x 2 A [ (B [ C) iff x 2 A or x 2 B [ C iff x 2 A or (x 2 B or x 2 C) iff x 2 A or x 2 B or x 2 C iff (x 2 A or x 2 B) or x 2 C iff x 2 A [ B or x 2 C iff x 2 (A [ B) [ C Definition 1.3 (Intersection). -
Mathematics 144 Set Theory Fall 2012 Version
MATHEMATICS 144 SET THEORY FALL 2012 VERSION Table of Contents I. General considerations.……………………………………………………………………………………………………….1 1. Overview of the course…………………………………………………………………………………………………1 2. Historical background and motivation………………………………………………………….………………4 3. Selected problems………………………………………………………………………………………………………13 I I. Basic concepts. ………………………………………………………………………………………………………………….15 1. Topics from logic…………………………………………………………………………………………………………16 2. Notation and first steps………………………………………………………………………………………………26 3. Simple examples…………………………………………………………………………………………………………30 I I I. Constructions in set theory.………………………………………………………………………………..……….34 1. Boolean algebra operations.……………………………………………………………………………………….34 2. Ordered pairs and Cartesian products……………………………………………………………………… ….40 3. Larger constructions………………………………………………………………………………………………..….42 4. A convenient assumption………………………………………………………………………………………… ….45 I V. Relations and functions ……………………………………………………………………………………………….49 1.Binary relations………………………………………………………………………………………………………… ….49 2. Partial and linear orderings……………………………..………………………………………………… ………… 56 3. Functions…………………………………………………………………………………………………………… ….…….. 61 4. Composite and inverse function.…………………………………………………………………………… …….. 70 5. Constructions involving functions ………………………………………………………………………… ……… 77 6. Order types……………………………………………………………………………………………………… …………… 80 i V. Number systems and set theory …………………………………………………………………………………. 84 1. The Natural Numbers and Integers…………………………………………………………………………….83 2. Finite induction -
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”. -
UNIT-I Mathematical Logic Statements and Notations
UNIT-I Mathematical Logic Statements and notations: A proposition or statement is a declarative sentence that is either true or false (but not both). For instance, the following are propositions: “Paris is in France” (true), “London is in Denmark” (false), “2 < 4” (true), “4 = 7 (false)”. However the following are not propositions: “what is your name?” (this is a question), “do your homework” (this is a command), “this sentence is false” (neither true nor false), “x is an even number” (it depends on what x represents), “Socrates” (it is not even a sentence). The truth or falsehood of a proposition is called its truth value. Connectives: Connectives are used for making compound propositions. The main ones are the following (p and q represent given propositions): Name Represented Meaning Negation ¬p “not p” Conjunction p q “p and q” Disjunction p q “p or q (or both)” Exclusive Or p q “either p or q, but not both” Implication p ⊕ q “if p then q” Biconditional p q “p if and only if q” Truth Tables: Logical identity Logical identity is an operation on one logical value, typically the value of a proposition that produces a value of true if its operand is true and a value of false if its operand is false. The truth table for the logical identity operator is as follows: Logical Identity p p T T F F Logical negation Logical negation is an operation on one logical value, typically the value of a proposition that produces a value of true if its operand is false and a value of false if its operand is true. -
6.5 the Recursion Theorem
6.5. THE RECURSION THEOREM 417 6.5 The Recursion Theorem The recursion Theorem, due to Kleene, is a fundamental result in recursion theory. Theorem 6.5.1 (Recursion Theorem, Version 1 )Letϕ0,ϕ1,... be any ac- ceptable indexing of the partial recursive functions. For every total recursive function f, there is some n such that ϕn = ϕf(n). The recursion Theorem can be strengthened as follows. Theorem 6.5.2 (Recursion Theorem, Version 2 )Letϕ0,ϕ1,... be any ac- ceptable indexing of the partial recursive functions. There is a total recursive function h such that for all x ∈ N,ifϕx is total, then ϕϕx(h(x)) = ϕh(x). 418 CHAPTER 6. ELEMENTARY RECURSIVE FUNCTION THEORY A third version of the recursion Theorem is given below. Theorem 6.5.3 (Recursion Theorem, Version 3 ) For all n ≥ 1, there is a total recursive function h of n +1 arguments, such that for all x ∈ N,ifϕx is a total recursive function of n +1arguments, then ϕϕx(h(x,x1,...,xn),x1,...,xn) = ϕh(x,x1,...,xn), for all x1,...,xn ∈ N. As a first application of the recursion theorem, we can show that there is an index n such that ϕn is the constant function with output n. Loosely speaking, ϕn prints its own name. Let f be the recursive function such that f(x, y)=x for all x, y ∈ N. 6.5. THE RECURSION THEOREM 419 By the s-m-n Theorem, there is a recursive function g such that ϕg(x)(y)=f(x, y)=x for all x, y ∈ N. -
April 22 7.1 Recursion Theorem
CSE 431 Theory of Computation Spring 2014 Lecture 7: April 22 Lecturer: James R. Lee Scribe: Eric Lei Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. They may be distributed outside this class only with the permission of the Instructor. An interesting question about Turing machines is whether they can reproduce themselves. A Turing machine cannot be be defined in terms of itself, but can it still somehow print its own source code? The answer to this question is yes, as we will see in the recursion theorem. Afterward we will see some applications of this result. 7.1 Recursion Theorem Our end goal is to have a Turing machine that prints its own source code and operates on it. Last lecture we proved the existence of a Turing machine called SELF that ignores its input and prints its source code. We construct a similar proof for the recursion theorem. We will also need the following lemma proved last lecture. ∗ ∗ Lemma 7.1 There exists a computable function q :Σ ! Σ such that q(w) = hPwi, where Pw is a Turing machine that prints w and hats. Theorem 7.2 (Recursion theorem) Let T be a Turing machine that computes a function t :Σ∗ × Σ∗ ! Σ∗. There exists a Turing machine R that computes a function r :Σ∗ ! Σ∗, where for every w, r(w) = t(hRi; w): The theorem says that for an arbitrary computable function t, there is a Turing machine R that computes t on hRi and some input. Proof: We construct a Turing Machine R in three parts, A, B, and T , where T is given by the statement of the theorem. -
Intuitionism on Gödel's First Incompleteness Theorem
The Yale Undergraduate Research Journal Volume 2 Issue 1 Spring 2021 Article 31 2021 Incomplete? Or Indefinite? Intuitionism on Gödel’s First Incompleteness Theorem Quinn Crawford Yale University Follow this and additional works at: https://elischolar.library.yale.edu/yurj Part of the Mathematics Commons, and the Philosophy Commons Recommended Citation Crawford, Quinn (2021) "Incomplete? Or Indefinite? Intuitionism on Gödel’s First Incompleteness Theorem," The Yale Undergraduate Research Journal: Vol. 2 : Iss. 1 , Article 31. Available at: https://elischolar.library.yale.edu/yurj/vol2/iss1/31 This Article is brought to you for free and open access by EliScholar – A Digital Platform for Scholarly Publishing at Yale. It has been accepted for inclusion in The Yale Undergraduate Research Journal by an authorized editor of EliScholar – A Digital Platform for Scholarly Publishing at Yale. For more information, please contact [email protected]. Incomplete? Or Indefinite? Intuitionism on Gödel’s First Incompleteness Theorem Cover Page Footnote Written for Professor Sun-Joo Shin’s course PHIL 437: Philosophy of Mathematics. This article is available in The Yale Undergraduate Research Journal: https://elischolar.library.yale.edu/yurj/vol2/iss1/ 31 Crawford: Intuitionism on Gödel’s First Incompleteness Theorem Crawford | Philosophy Incomplete? Or Indefinite? Intuitionism on Gödel’s First Incompleteness Theorem By Quinn Crawford1 1Department of Philosophy, Yale University ABSTRACT This paper analyzes two natural-looking arguments that seek to leverage Gödel’s first incompleteness theo- rem for and against intuitionism, concluding in both cases that the argument is unsound because it equivo- cates on the meaning of “proof,” which differs between formalism and intuitionism. -
Introduction to Logic
IntroductionI Introduction to Logic Propositional calculus (or logic) is the study of the logical Christopher M. Bourke relationship between objects called propositions and forms the basis of all mathematical reasoning. Definition A proposition is a statement that is either true or false, but not both (we usually denote a proposition by letters; p, q, r, s, . .). Computer Science & Engineering 235 Introduction to Discrete Mathematics [email protected] IntroductionII ExamplesI Example (Propositions) Definition I 2 + 2 = 4 I The derivative of sin x is cos x. The value of a proposition is called its truth value; denoted by T or I 6 has 2 factors 1 if it is true and F or 0 if it is false. Opinions, interrogative and imperative sentences are not Example (Not Propositions) propositions. I C++ is the best language. I When is the pretest? I Do your homework. ExamplesII Logical Connectives Connectives are used to create a compound proposition from two Example (Propositions?) or more other propositions. I Negation (denoted or !) ¬ I 2 + 2 = 5 I And (denoted ) or Logical Conjunction ∧ I Every integer is divisible by 12. I Or (denoted ) or Logical Disjunction ∨ I Microsoft is an excellent company. I Exclusive Or (XOR, denoted ) ⊕ I Implication (denoted ) → I Biconditional; “if and only if” (denoted ) ↔ Negation Logical And A proposition can be negated. This is also a proposition. We usually denote the negation of a proposition p by p. ¬ The logical connective And is true only if both of the propositions are true. It is also referred to as a conjunction. Example (Negated Propositions) Example (Logical Connective: And) I Today is not Monday. -
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. -
Folktale Documentation Release 1.0
Folktale Documentation Release 1.0 Quildreen Motta Nov 05, 2017 Contents 1 Guides 3 2 Indices and tables 5 3 Other resources 7 3.1 Getting started..............................................7 3.2 Folktale by Example........................................... 10 3.3 API Reference.............................................. 12 3.4 How do I................................................... 63 3.5 Glossary................................................. 63 Python Module Index 65 i ii Folktale Documentation, Release 1.0 Folktale is a suite of libraries for generic functional programming in JavaScript that allows you to write elegant modular applications with fewer bugs, and more reuse. Contents 1 Folktale Documentation, Release 1.0 2 Contents CHAPTER 1 Guides • Getting Started A series of quick tutorials to get you up and running quickly with the Folktale libraries. • API reference A quick reference of Folktale’s libraries, including usage examples and cross-references. 3 Folktale Documentation, Release 1.0 4 Chapter 1. Guides CHAPTER 2 Indices and tables • Global Module Index Quick access to all modules. • General Index All functions, classes, terms, sections. • Search page Search this documentation. 5 Folktale Documentation, Release 1.0 6 Chapter 2. Indices and tables CHAPTER 3 Other resources • Licence information 3.1 Getting started This guide will cover everything you need to start using the Folktale project right away, from giving you a brief overview of the project, to installing it, to creating a simple example. Once you get the hang of things, the Folktale By Example guide should help you understanding the concepts behind the library, and mapping them to real use cases. 3.1.1 So, what’s Folktale anyways? Folktale is a suite of libraries for allowing a particular style of functional programming in JavaScript. -
Theorem Proving in Classical Logic
MEng Individual Project Imperial College London Department of Computing Theorem Proving in Classical Logic Supervisor: Dr. Steffen van Bakel Author: David Davies Second Marker: Dr. Nicolas Wu June 16, 2021 Abstract It is well known that functional programming and logic are deeply intertwined. This has led to many systems capable of expressing both propositional and first order logic, that also operate as well-typed programs. What currently ties popular theorem provers together is their basis in intuitionistic logic, where one cannot prove the law of the excluded middle, ‘A A’ – that any proposition is either true or false. In classical logic this notion is provable, and the_: corresponding programs turn out to be those with control operators. In this report, we explore and expand upon the research about calculi that correspond with classical logic; and the problems that occur for those relating to first order logic. To see how these calculi behave in practice, we develop and implement functional languages for propositional and first order logic, expressing classical calculi in the setting of a theorem prover, much like Agda and Coq. In the first order language, users are able to define inductive data and record types; importantly, they are able to write computable programs that have a correspondence with classical propositions. Acknowledgements I would like to thank Steffen van Bakel, my supervisor, for his support throughout this project and helping find a topic of study based on my interests, for which I am incredibly grateful. His insight and advice have been invaluable. I would also like to thank my second marker, Nicolas Wu, for introducing me to the world of dependent types, and suggesting useful resources that have aided me greatly during this report. -
John P. Burgess Department of Philosophy Princeton University Princeton, NJ 08544-1006, USA [email protected]
John P. Burgess Department of Philosophy Princeton University Princeton, NJ 08544-1006, USA [email protected] LOGIC & PHILOSOPHICAL METHODOLOGY Introduction For present purposes “logic” will be understood to mean the subject whose development is described in Kneale & Kneale [1961] and of which a concise history is given in Scholz [1961]. As the terminological discussion at the beginning of the latter reference makes clear, this subject has at different times been known by different names, “analytics” and “organon” and “dialectic”, while inversely the name “logic” has at different times been applied much more broadly and loosely than it will be here. At certain times and in certain places — perhaps especially in Germany from the days of Kant through the days of Hegel — the label has come to be used so very broadly and loosely as to threaten to take in nearly the whole of metaphysics and epistemology. Logic in our sense has often been distinguished from “logic” in other, sometimes unmanageably broad and loose, senses by adding the adjectives “formal” or “deductive”. The scope of the art and science of logic, once one gets beyond elementary logic of the kind covered in introductory textbooks, is indicated by two other standard references, the Handbooks of mathematical and philosophical logic, Barwise [1977] and Gabbay & Guenthner [1983-89], though the latter includes also parts that are identified as applications of logic rather than logic proper. The term “philosophical logic” as currently used, for instance, in the Journal of Philosophical Logic, is a near-synonym for “nonclassical logic”. There is an older use of the term as a near-synonym for “philosophy of language”.