The Big Proof Agenda=1 Supported by SRI International

Total Page:16

File Type:pdf, Size:1020Kb

The Big Proof Agenda=1 Supported by SRI International The Big Proof Agenda1 N. Shankar Computer Science Laboratory SRI International Menlo Park, CA June 26, 2017 1 Supported by SRI International. Proofs and Machines In science nothing capable of proof ought to be accepted without proof. |Richard Dedekind The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so one can prove any theorem us- ing nothing but a few mechanical rules. {Kurt G¨odel Natarajan Shankar Pragmatics of Formal Proof 2/33 Proofs and Machines Checking mathematical proofs is potentially one of the most interesting and useful applications of automatic computers. |John McCarthy Science is what we understand well enough to explain to a computer. Art is everything else we do. |Donald Knuth Natarajan Shankar Pragmatics of Formal Proof 3/33 Overview Mathematics is the study of abstractions such as collections, maps, graphs, algebras, sequences, etc. It has been \unreasonably effective” as a language and a foundation for a growing number of other disciplines. Proofs are used to rigorously derive irrefutable conclusions using a small number of `self-evident' principles of reasoning. Formal proofs can be constructed from a small number of precisely stated axioms and rules of inference. Proof Assistants, Decision Procedures, And Theorem Provers make it possible to construct formal proofs and formally verify mathematical claims. The goal of the INI Big Proof program is to define the future direction of proof technology so that it is used by Mathematicians to discover and verify new results Educators to communicate proofs and problems solving techniques Natarajan Shankar Pragmatics of Formal Proof 4/33 In the Beginning Thales of Miletus (624 { 547 B.C.): Earliest known person to be credited with theorems and proofs. It was Thales who first conceived the principle of explaining the multitude of phenomena by a small number of hy- potheses for all the various manifesta- tions of matter. Pythagoras of Samos (569{475 B.C.): Sys- tematic study of mathematics for its own sake. ... he tried to use his symbolic method of teaching which was similar in all respects to the lessons he had learnt in Egypt. The Samians were not very keen on this method and treated him in a rude and improper manner. Natarajan Shankar Pragmatics of Formal Proof 5/33 The Axiomatic Method Plato (427{347 B.C.): Suggested the idea of a single axiom system for all knowledge. the reality which scientific thought is seek- ing must be expressible in mathematical terms, mathematics being the most pre- cise and definite kind of thinking of which we are capable. Aristotle (384{322 B.C.) of Stagira: Laid the foun- dation for scientific thought by proposing that all theoretical disciplines must be based on axiomatic principles. Natarajan Shankar Pragmatics of Formal Proof 6/33 Euclid and The Elements Euclid of Alexandria (325{265 B.C.): Systematic compila- tion and exposition of geometry and number theory. 1 A straight line segment can be drawn joining any two points. 2 Any straight line segment can be extended indefinitely in a straight line. 3 Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4 All right angles are congruent. 5 If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate. Natarajan Shankar Pragmatics of Formal Proof 7/33 Euclid's Pons Asinorum [Wikipedia] Charles Dodgson, quoted in Ian Stewart, Concepts of Modern Mathematics: MINOS: It is proposed to prove [the theorem] by taking up the isosceles triangle, turning it over, and then lay- ing it down again upon itself. EUCLID: Surely that has too much of the Irish Bull about it, and reminds one a little too vividly of the man who walked down his own throat, to de- serve a place in a strictly philosophi- cal treatise? MINOS: I suppose its defenders would say that it is conceived to leave a trace of itself behind and the re- versed triangle is laid down on the trace so left. Natarajan Shankar Pragmatics of Formal Proof 8/33 Mechanizing Reasoning: Llull and Leibniz Ramon Llull (1235{1316): Talked of reducing all knowl- edge to first principles. Developed a symbolic notation (Ars Magna) and conceived of a reasoning machine. When he attempted to apply rational thinking to religion, Pope Gregor XI \accused him of confusing faith with reason and condemned his teachings." Gottfried Leibniz (1646{1716) The idea of a formal lan- guage (characteristica universalis) for expressing scholarly knowledge and a mechanical method for making deductions (calculus ratiocinator). What must be achieved is in fact this: that every paralogism be recognized as an error of calculation, and every sophism when expressed in this new kind of notation, appear as a solecism or barbarism, to be corrected easily by the laws of this philosophical grammar. Once this is done, then when a controversy arises, disputa- tion will no more be needed between two philosophers than between two computers. It will suffice that, pen in hand, they sit down to their abacus and (calling in a friend, if they so wish) say to each other: let us calculate. Natarajan Shankar Pragmatics of Formal Proof 9/33 Formal Arithmetic Richard Dedekind (1813{1916) : Early set theoretic ideas, infinite sets, Dedekind cuts, arithmetic. Giuseppe Peano (1858{1932) : Modern logic notation, arithmetic. 0 6= S(x) S(x) = S(y) =) x = y A(0) ^ (8x:A(x) =) A(S(x))) =) (8x:A(x)) Natarajan Shankar Pragmatics of Formal Proof 10/33 Set Theory Georg Cantor (1845{1918): Set theory, transfinite sets, continuum hypothesis. Ernst Zermelo (1871{1953): For- malized set theory, axiom of choice, well-ordering principle. Extensionality: x = y () (8z:z 2 x () z 2 y) Empty set 8x::x 2 ; Pairing: 8x; y:9z:8u:u 2 z () u = x _ u = y Union: 8x:9y:8z:z 2 y () (9w:z 2 w ^ w 2 x) Separation: fx 2 yjAg, for any formula A, y 62 vars(A). Infinity: There is a set containing all the finite ordinals. Power set: For any set, we have the set of all its subsets. Natarajan Shankar Pragmatics of Formal Proof 11/33 Modern Formal Logic Gottlob Frege (1848{1925): A system of quantifi- cational logic. Bertrand Russell (1872{1970) and Al- fred North Whitehead (1861{1947): Ramified type theory, rigorous formal development of a significant portion of mathematics. ()i is the type of propositions of order i. i1 im i (τ1 ; : : : ; τm ) with i > max(i1;:::; in), is the type of propositional functions from τ1 × ::: × τm, possibly quantifying over variables of order below i. Natarajan Shankar Pragmatics of Formal Proof 12/33 Simple Type Theory Kurt G¨odel(1906{1978): Completeness: Every statement has a counter-model or a proof. Incompleteness: Any consistent formal theory for arithmetic contains statements that are nei- ther provable nor disprovable. Such a theory cannot prove its own consistency. Alonzo Church (1909{1995): Lambda calculus, recursive unsolvability, Church's thesis, simple type theory. The types of individuals i and propositions o at level 0 The function type A!B is a type at level n + 1 if A is a type at level at most n and B is a type at level at most n + 1. Natarajan Shankar Pragmatics of Formal Proof 13/33 Computing Machinery and Intelligence Turing Machine (TM) as a finite state machine scanning and writing an infinite tape. A Universal Turing Machine (UTM) as an interpreter for a given TM on a given input. The unsolvability of the Halting Problem: Does a TM M diverge on an input i? The unsolvability of the Entscheidungsproblem: Is a given sentence in first-order logic valid? Natarajan Shankar Pragmatics of Formal Proof 14/33 The First Automated Reasoner In 1954, Martin Davis programmed a Presburger Arithmetic decision procedure. Its great triumph was to prove that the sum of two even numbers is even. Natarajan Shankar Pragmatics of Formal Proof 15/33 Early Proof Assistants [Automath, LCF, Thm/Nqthm/ACL2] John McCarthy N. G. de Bruijn Woody Bledsoe Robin Milner Bob Boyer J Moore Natarajan Shankar Pragmatics of Formal Proof 16/33 The Language of Formal Mathematics: Recursion ACL2 (1972{present) is based on Primitive Recursive Arithmetic (adapted to Lisp) and is Adequate for formalizing combinatorial mathematics. Supported by impressive mechanization in the form of induction, simplification, and rewriting. (DEFUN REV (X) (IF (ENDP X) NIL (APPEND (REV (CDR X)) (LIST (CAR X))))) (DEFTHM REV_OF_REV (IMPLIES (TRUE-LISTP X) (EQUAL (REV (REV X)) X))) Natarajan Shankar Pragmatics of Formal Proof 17/33 The Language of Formal Mathematics: Higher-Order Logic Church's introduced a simple theory of types consisting of Types: bool, nat, A!B Terms: x, c. =A, λx:a, f a. Peter Andrews in the 1970s used higher-order logic in his theorem prover TPS. HOL, HOL-Light, and Isabelle/HOL use classical higher-order logic based on Church's simple theory of types which Can be interpreted in Zermelo set theory. Captures basic mathematical concepts adequately. Natarajan Shankar Pragmatics of Formal Proof 18/33 The Language of Formal Mathematics: Dependent Higher-Order Logic PVS extends higher-order logic with subtypes (e.g., primes, continuity-preserving operators) and dependent types (e.g., finite sequences) Proof obligations are generated to check that an expression has the expected type. Type checking is undecidable since proof obligations can be arbitrary formulas.
Recommended publications
  • “The Church-Turing “Thesis” As a Special Corollary of Gödel's
    “The Church-Turing “Thesis” as a Special Corollary of Gödel’s Completeness Theorem,” in Computability: Turing, Gödel, Church, and Beyond, B. J. Copeland, C. Posy, and O. Shagrir (eds.), MIT Press (Cambridge), 2013, pp. 77-104. Saul A. Kripke This is the published version of the book chapter indicated above, which can be obtained from the publisher at https://mitpress.mit.edu/books/computability. It is reproduced here by permission of the publisher who holds the copyright. © The MIT Press The Church-Turing “ Thesis ” as a Special Corollary of G ö del ’ s 4 Completeness Theorem 1 Saul A. Kripke Traditionally, many writers, following Kleene (1952) , thought of the Church-Turing thesis as unprovable by its nature but having various strong arguments in its favor, including Turing ’ s analysis of human computation. More recently, the beauty, power, and obvious fundamental importance of this analysis — what Turing (1936) calls “ argument I ” — has led some writers to give an almost exclusive emphasis on this argument as the unique justification for the Church-Turing thesis. In this chapter I advocate an alternative justification, essentially presupposed by Turing himself in what he calls “ argument II. ” The idea is that computation is a special form of math- ematical deduction. Assuming the steps of the deduction can be stated in a first- order language, the Church-Turing thesis follows as a special case of G ö del ’ s completeness theorem (first-order algorithm theorem). I propose this idea as an alternative foundation for the Church-Turing thesis, both for human and machine computation. Clearly the relevant assumptions are justified for computations pres- ently known.
    [Show full text]
  • A One-Page Primer the Centers and Circles of a Triangle
    1 The Triangle: A One-Page Primer (Date: March 15, 2004 ) An angle is formed by two intersecting lines. A right angle has 90◦. An angle that is less (greater) than a right angle is called acute (obtuse). Two angles whose sum is 90◦ (180◦) are called complementary (supplementary). Two intersecting perpendicular lines form right angles. The angle bisectors are given by the locus of points equidistant from its two lines. The internal and external angle bisectors are perpendicular to each other. The vertically opposite angles formed by the intersection of two lines are equal. In triangle ABC, the angles are denoted A, B, C; the sides are denoted a = BC, b = AC, c = AB; the semi- 1 ◦ perimeter (s) is 2 the perimeter (a + b + c). A + B + C = 180 . Similar figures are equiangular and their corresponding components are proportional. Two triangles are similar if they share two equal angles. Euclid's VI.19, 2 ∆ABC a the areas of similar triangles are proportional to the second power of their corresponding sides ( ∆A0B0C0 = a02 ). An acute triangle has three acute angles. An obtuse triangle has one obtuse angle. A scalene triangle has all its sides unequal. An isosceles triangle has two equal sides. I.5 (pons asinorum): The angles at the base of an isosceles triangle are equal. An equilateral triangle has all its sides congruent. A right triangle has one right angle (at C). The hypotenuse of a right triangle is the side opposite the 90◦ angle and the catheti are its other legs. In a opp adj opp adj hyp hyp right triangle, sin = , cos = , tan = , cot = , sec = , csc = .
    [Show full text]
  • The Cyber Entscheidungsproblem: Or Why Cyber Can’T Be Secured and What Military Forces Should Do About It
    THE CYBER ENTSCHEIDUNGSPROBLEM: OR WHY CYBER CAN’T BE SECURED AND WHAT MILITARY FORCES SHOULD DO ABOUT IT Maj F.J.A. Lauzier JCSP 41 PCEMI 41 Exercise Solo Flight Exercice Solo Flight Disclaimer Avertissement Opinions expressed remain those of the author and Les opinons exprimées n’engagent que leurs auteurs do not represent Department of National Defence or et ne reflètent aucunement des politiques du Canadian Forces policy. This paper may not be used Ministère de la Défense nationale ou des Forces without written permission. canadiennes. Ce papier ne peut être reproduit sans autorisation écrite. © Her Majesty the Queen in Right of Canada, as © Sa Majesté la Reine du Chef du Canada, représentée par represented by the Minister of National Defence, 2015. le ministre de la Défense nationale, 2015. CANADIAN FORCES COLLEGE – COLLÈGE DES FORCES CANADIENNES JCSP 41 – PCEMI 41 2014 – 2015 EXERCISE SOLO FLIGHT – EXERCICE SOLO FLIGHT THE CYBER ENTSCHEIDUNGSPROBLEM: OR WHY CYBER CAN’T BE SECURED AND WHAT MILITARY FORCES SHOULD DO ABOUT IT Maj F.J.A. Lauzier “This paper was written by a student “La présente étude a été rédigée par un attending the Canadian Forces College stagiaire du Collège des Forces in fulfilment of one of the requirements canadiennes pour satisfaire à l'une des of the Course of Studies. The paper is a exigences du cours. L'étude est un scholastic document, and thus contains document qui se rapporte au cours et facts and opinions, which the author contient donc des faits et des opinions alone considered appropriate and que seul l'auteur considère appropriés et correct for the subject.
    [Show full text]
  • Read Book Advanced Euclidean Geometry Ebook
    ADVANCED EUCLIDEAN GEOMETRY PDF, EPUB, EBOOK Roger A. Johnson | 336 pages | 30 Nov 2007 | Dover Publications Inc. | 9780486462370 | English | New York, United States Advanced Euclidean Geometry PDF Book As P approaches nearer to A , r passes through all values from one to zero; as P passes through A , and moves toward B, r becomes zero and then passes through all negative values, becoming —1 at the mid-point of AB. Uh-oh, it looks like your Internet Explorer is out of date. In Elements Angle bisector theorem Exterior angle theorem Euclidean algorithm Euclid's theorem Geometric mean theorem Greek geometric algebra Hinge theorem Inscribed angle theorem Intercept theorem Pons asinorum Pythagorean theorem Thales's theorem Theorem of the gnomon. It might also be so named because of the geometrical figure's resemblance to a steep bridge that only a sure-footed donkey could cross. Calculus Real analysis Complex analysis Differential equations Functional analysis Harmonic analysis. This article needs attention from an expert in mathematics. Facebook Twitter. On any line there is one and only one point at infinity. This may be formulated and proved algebraically:. When we have occasion to deal with a geometric quantity that may be regarded as measurable in either of two directions, it is often convenient to regard measurements in one of these directions as positive, the other as negative. Logical questions thus become completely independent of empirical or psychological questions For example, proposition I. This volume serves as an extension of high school-level studies of geometry and algebra, and He was formerly professor of mathematics education and dean of the School of Education at The City College of the City University of New York, where he spent the previous 40 years.
    [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]
  • Geometry by Its History
    Undergraduate Texts in Mathematics Geometry by Its History Bearbeitet von Alexander Ostermann, Gerhard Wanner 1. Auflage 2012. Buch. xii, 440 S. Hardcover ISBN 978 3 642 29162 3 Format (B x L): 15,5 x 23,5 cm Gewicht: 836 g Weitere Fachgebiete > Mathematik > Geometrie > Elementare Geometrie: Allgemeines Zu Inhaltsverzeichnis schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, eBooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte. 2 The Elements of Euclid “At age eleven, I began Euclid, with my brother as my tutor. This was one of the greatest events of my life, as dazzling as first love. I had not imagined that there was anything as delicious in the world.” (B. Russell, quoted from K. Hoechsmann, Editorial, π in the Sky, Issue 9, Dec. 2005. A few paragraphs later K. H. added: An innocent look at a page of contemporary the- orems is no doubt less likely to evoke feelings of “first love”.) “At the age of 16, Abel’s genius suddenly became apparent. Mr. Holmbo¨e, then professor in his school, gave him private lessons. Having quickly absorbed the Elements, he went through the In- troductio and the Institutiones calculi differentialis and integralis of Euler. From here on, he progressed alone.” (Obituary for Abel by Crelle, J. Reine Angew. Math. 4 (1829) p. 402; transl. from the French) “The year 1868 must be characterised as [Sophus Lie’s] break- through year.
    [Show full text]
  • Euclid Transcript
    Euclid Transcript Date: Wednesday, 16 January 2002 - 12:00AM Location: Barnard's Inn Hall Euclid Professor Robin Wilson In this sequence of lectures I want to look at three great mathematicians that may or may not be familiar to you. We all know the story of Isaac Newton and the apple, but why was it so important? What was his major work (the Principia Mathematica) about, what difficulties did it raise, and how were they resolved? Was Newton really the first to discover the calculus, and what else did he do? That is next time a fortnight today and in the third lecture I will be talking about Leonhard Euler, the most prolific mathematician of all time. Well-known to mathematicians, yet largely unknown to anyone else even though he is the mathematical equivalent of the Enlightenment composer Haydn. Today, in the immortal words of Casablanca, here's looking at Euclid author of the Elements, the best-selling mathematics book of all time, having been used for more than 2000 years indeed, it is quite possibly the most printed book ever, apart than the Bible. Who was Euclid, and why did his writings have such influence? What does the Elements contain, and why did one feature of it cause so much difficulty over the years? Much has been written about the Elements. As the philosopher Immanuel Kant observed: There is no book in metaphysics such as we have in mathematics. If you want to know what mathematics is, just look at Euclid's Elements. The Victorian mathematician Augustus De Morgan, of whom I told you last October, said: The thirteen books of Euclid must have been a tremendous advance, probably even greater than that contained in the Principia of Newton.
    [Show full text]
  • Lambda Calculus and the Decision Problem
    Lambda Calculus and the Decision Problem William Gunther November 8, 2010 Abstract In 1928, Alonzo Church formalized the λ-calculus, which was an attempt at providing a foundation for mathematics. At the heart of this system was the idea of function abstraction and application. In this talk, I will outline the early history and motivation for λ-calculus, especially in recursion theory. I will discuss the basic syntax and the primary rule: β-reduction. Then I will discuss how one would represent certain structures (numbers, pairs, boolean values) in this system. This will lead into some theorems about fixed points which will allow us have some fun and (if there is time) to supply a negative answer to Hilbert's Entscheidungsproblem: is there an effective way to give a yes or no answer to any mathematical statement. 1 History In 1928 Alonzo Church began work on his system. There were two goals of this system: to investigate the notion of 'function' and to create a powerful logical system sufficient for the foundation of mathematics. His main logical system was later (1935) found to be inconsistent by his students Stephen Kleene and J.B.Rosser, but the 'pure' system was proven consistent in 1936 with the Church-Rosser Theorem (which more details about will be discussed later). An application of λ-calculus is in the area of computability theory. There are three main notions of com- putability: Hebrand-G¨odel-Kleenerecursive functions, Turing Machines, and λ-definable functions. Church's Thesis (1935) states that λ-definable functions are exactly the functions that can be “effectively computed," in an intuitive way.
    [Show full text]
  • A Rewrite System for Elimination of the Functional Extensionality and Univalence Axioms
    A Rewrite System for Elimination of the Functional Extensionality and Univalence Axioms Raphaël Cauderlier Abstract We explore the adaptation of the ideas that we developed for automatic elimination of classical axioms [5] to extensionality axioms in Type Theory. Extensionality axioms are added to Type Theory when we lack a way to prove equality between two indistinguishable objects. These indistinguishable objects might be logically-equivalent propo- sitions, pointwise equal functions, pointwise equivalent predicates, or equivalent (isomorphic) types leading to the various following extensionality axioms: Propositional extensionality: PE :ΠP; Q : Prop: (P , Q) ! P =Prop Q Non-dependent functional extensionality: NFE :ΠA; B : U: Πf; g :(A ! B): (Πa : A: f a =B g a) ! f =A!B g Dependent functional extensionality: DF E :ΠA : U: ΠB :(A ! U): Πf; g : (Πa : A ! B(a)): (Πa : A: f a =B(a) g a) ! f =Πa:A: B(a) g Predicate extensionality: P redE :ΠA : U: ΠP; Q :(A ! Prop): (Πa : A: P a , Q a) ! P =A!Prop Q Set extensionality: SetE :ΠA : U: ΠX; Y : set(A): (Πa : A: a 2 X , a 2 Y ) ! X =set(A) Y Univalence: let idtoeqv be the canonical function of type ΠA; B : U: (A =U B) ! (A ≈ B), for all types A; B : U, idtoeqv AB is an equivalence between the types A =U B and A ≈ B. In these axiom statements, the symbol =A denotes equality between two terms of type A, set(A) is type of the sets of elements of A, 2 is the set membership relation, U is a universe and ' is the equivalence relation over types in U.
    [Show full text]
  • The Axiom of Choice and the Law of Excluded Middle in Weak Set Theories
    The Axiom of Choice and the Law of Excluded Middle in Weak Set Theories John L. Bell Department of Philosophy, University of Western Ontario In constructive mathematics the axiom of choice (AC) has a somewhat ambiguous status. On the one hand, in intuitionistic set theory, or the local set theory associated with a topos ([2]) it can be shown to entail the law of excluded middle (LEM) ([ 3 ], [ 5 ]). On the other hand, under the “propositions-as types” interpretation which lies at the heart of constructive predicative type theories such as that of Martin-Löf [9], the axiom of choice is actually derivable (see, e.g. [11] ), and so certainly cannot entail the law of excluded middle. This incongruity has been the subject of a number of recent investigations, for example [6], [7], [9], [12]. What has emerged is that for the derivation of LEM from AC to go through it is sufficient that sets (in particular power sets), or functions, have a degree of extensionality which is, so to speak, built into the usual set theories but is incompatible with constructive type theories Another condition, independent of extensionality, ensuring that the derivation goes through is that any equivalence relation determines a quotient set. LEM can also be shown to follow from a suitably extensionalized version of AC. The arguments establishing these intriguing results have mostly been formulated within a type-theoretic framework. It is my purpose here to formulate and derive analogous results within a comparatively straightforward set-theoretic framework. The core principles of this framework form a theory – weak set theory WST – which lacks the axiom of extensionality1 and supports only minimal set-theoretic constructions.
    [Show full text]
  • Does Universalism Entail Extensionalism?
    NOUSˆ 00:00 (2014) 1–12 doi: 10.1111/nous.12063 Does Universalism Entail Extensionalism? A. J. COTNOIR University of St Andrews Abstract Does a commitment to mereological universalism automatically bring along a commitment to the controversial doctrine of mereological extensionalism—the view that objects with the same proper parts are identical? A recent argument suggests the answer is ‘yes’. This paper attempts a systematic response to the argument, considering nearly every available line of reply. It argues that only one approach— the mutual parts view—can yield a viable mereology where universalism does not entail extensionalism. Here is an axiomatization of classical extensional mereology, where ‘<’ stands for the proper parthood relation, ‘≤’ stands for the parthood relation (i.e. ‘x ≤ y’ abbreviates ‘x < y ∨ x = y’), and ◦ for mereological overlap (i.e. ‘x ◦ y’abbreviates ‘∃z(z ≤ x ∧ z ≤ y)’). Transitivity (x < y ∧ y < z) → x < z Weak Supplementation x < y →∃z(z ≤ y ∧¬z ◦ x)) Unrestricted Fusion ∀xx∃y F(y, xx). Where fusions are defined via: Fusion F(t, xx):= xx ≤ t ∧∀y(y ≤ t → y ◦ xx)) To simplify notation, I use ‘xx ≤ y’ to mean that each x among xx is part of y; likewise ‘y ◦ xx’meansthaty overlaps some x among xx. It is worth noting that weak supplementation entails the remaining (strict) partial order axioms:1 Irreflexivity x < x Asymmetry x < y → y < x The definition of ≤ provides the corresponding weak partial order principles of reflexivity and antisymmetry. Listing these as separate axioms is unnecessary here; however, this will not be so in what follows.
    [Show full text]