DOCSLIB.ORG

Mathematical Logic and Computability

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mathematical Logic and Computability Mathematical Logic and Computability J. Keisler, K. Kunen, T. Millar, A. Miller, J. Robbin February 10, 2006 This version is from Spring 1987 0 Contents 1 Propositional Logic 5 1.1 Syntax of Propositional Logic. 8 1.2 Semantics of Propositional Logic. 10 1.3 Truth Tables . 13 1.4 Induction on formulas and unique readability . 14 1.5 Tableaus. 17 1.6 Soundness and Confutation . 21 1.7 Completeness . 23 1.8 Computer Problem . 29 1.9 Exercises . 31 2 Pure Predicate Logic 37 2.1 Syntax of Predicate Logic. 40 2.2 Free and Bound Variables. 42 2.3 Semantics of Predicate Logic. 44 2.4 A simpler notation. 49 2.5 Tableaus. 51 2.6 Soundness . 55 2.7 Completeness . 56 2.8 Computer problem using PREDCALC . 60 2.9 Computer problem . 61 2.10 Exercises . 64 3 Full Predicate Logic 68 3.1 Semantics. 70 3.2 Tableaus. 72 3.3 Soundness . 75 3.4 Completeness . 76 3.5 Peano Arithmetic . 79 3.6 Computer problem . 80 3.7 Exercises . 82 4 Computable Functions 87 4.1 Numerical Functions. 87 4.2 Examples. 87 1 4.3 Extension. 89 4.4 Numerical Relations. 90 4.5 The Unlimited Register Machine. 90 4.6 Examples of URM-Computable Functions. 92 4.7 Universal URM machines . 109 4.8 Extract and Put . 115 4.9 UNIV.GN commented listing . 121 4.10 Primitive Recursion. 126 4.11 Recursive functions . 128 4.12 The URM-Computable Functions are recursive. 132 4.13 Decidable Relations. 138 4.14 Partial decidability . 138 4.15 The Diagonal Method . 138 4.16 The Halting Problem. 140 4.17 The G¨odel Incompleteness Theorem . 140 4.18 The Undecidability of Predicate Logic. 142 4.19 Computer problem . 143 4.20 Computer problem . 143 4.21 Exercises . 146 5 Mathematical Lingo 149 5.1 Sets. 149 5.2 Functions. 152 5.3 Inverses. 154 5.4 Cartesian Product. 158 5.5 Set theoretic operations. 160 5.6 Finite Sets . 163 5.7 Equivalence Relations. 164 5.8 Induction on the Natural Numbers . 166 6 Computer program documentation 169 6.1 TABLEAU program documentation . 169 6.2 COMPLETE program documentation . 179 6.3 PREDCALC program documentation . 179 6.4 BUILD program documentation . 186 6.5 MODEL program documentation . 189 6.6 GNUMBER program documentation . 190 2 A A Simple Proof. 199 B A lemma on valuations. 199 C Summary of Syntax Rules 204 D Summary of Tableaus. 207 E Finished Sets. 209 F Commented outline of the PARAM program 211 G Index 213 List of Figures 1 Propositional Extension Rules. 22 2 Proof of Extension Lemma . 26 3 Quantifier Rules for Pure Predicate Logic. 53 4 flowchart for add . 94 5 ADD.GN . 95 6 flowchart for multiplication . 96 7 assembly program for mult . 97 8 MULT.GN . 98 9 Psuedocode for Predescessor . 99 10 flowchart for predescessor . 100 11 PRED.GN . 101 12 flowchart for dotminus . 102 13 dotminus assembly code . 103 14 DOTMINUS.GN . 104 15 divide with remainder (divrem) . 105 16 expanding part of the flowchart for divrem . 106 17 divrem assembly code . 107 18 DIVREM.GN . 108 19 flowchart for univ . 112 20 flowchart for nextstate . 113 21 Jump: magnification of J box . 114 3 22 UNIV2.GN part1 . 123 23 UNIV.GN part 2 . 124 24 UNIV.GN initialization routine . 125 25 Hypothesis Mode . 171 26 Tableau Mode . 174 27 Map Mode . 178 28 Help box for the & key . 183 29 Help box for the R(...) key . 183 4 1 Propositional Logic In this book we shall study certain formal languages each of which abstracts from ordinary mathematical language (and to a lesser extent, everyday En- glish) some aspects of its logical structure. Formal languages differ from natural languages such as English in that the syntax of a formal language is precisely given. This is not the case with English: authorities often disagree as to whether a given English sentence is grammatically correct. Mathe- matical logic may be defined as that branch of mathematics which studies formal languages. In this chapter we study a formal language called propositional logic. This language abstracts from ordinary language the properties of the propo- sitional connectives (commonly called conjunctions by grammarians). These are “not”, “and”, “or”, “if”, and “if and only if”. These connectives are ex- tensional in the sense that the truth value of a compound sentence built up from these connectives depends only the truth values of the component sim- ple sentences. (The conjunction “because” is not extensional in this sense: one can easily give examples of English sentences p, q, and r such that p, q, r are true, but ‘p because q’ is true and ‘p because r’ is false.) Further- more, we are only concerned with the meanings that common mathematical usage accord these connectives; this is sometimes slightly different from their meanings in everyday English. We now explain these meanings. NEGATION. A sentence of form ‘not p’ is true exactly when p is false. The symbol used in mathematical logic for “not” is ¬ (but in older books the symbol ∼ was used). Thus of the two sentences ¬2 + 2 = 4 ¬2 + 2 = 5 the first is false while the second is true. The sentence ¬p is called the negation of p. CONJUNCTION. A sentence of form ‘p and q’ is true exactly when both p and q are true. The mathematical symbol for “and” is ∧ (or & in some older books). Thus of the four sentences 2 + 2 = 4 ∧ 2 + 3 = 5 5 2 + 2 = 4 ∧ 2 + 3 = 7 2 + 2 = 6 ∧ 2 + 3 = 5 2 + 2 = 6 ∧ 2 + 3 = 7 the first is true and the last three are false. The sentence p ∧ q is called the conjunction of p and q. For the mathematician, the words “and” and “but” have the same mean- ing. In everyday English these words cannot be used interchangeably, but the difference is psychological rather than logical. DISJUNCTION. A sentence of form ‘p or q’ is true exactly when either p is true or q is true (or both). The symbol use in mathematical logic for “or” is ∨. Thus of the four sentences 2 + 2 = 4 ∨ 2 + 3 = 5 2 + 2 = 4 ∨ 2 + 3 = 7 2 + 2 = 6 ∨ 2 + 3 = 5 2 + 2 = 6 ∨ 2 + 3 = 7 the first three are true while the last is false. The sentence p ∨ q is called the disjunction of p and q. Occasionally, the sentence p or q has a different meaning in everyday life from that just given. For example, the phrase “soup or salad included” in a restaurant menu means that the customer can have either soup or salad with his/her dinner at no extra cost but not both. This usage of the word “or” is called exclusive (because it excludes the case where both components are true). Mathematicians generally use the inclusive meaning explained above; when they intend the exclusive meaning they say so explicitly as in p or q but not both. IMPLICATION. The forms ‘if p, then q’, ‘q, if p’, ‘p implies q’, ‘p only if q’, and ‘q whenever p’ all having the same meaning for the mathematician: p ‘implies q’ is false exactly when p is true but q is false. The mathematical symbol for “implies” is ⇒ (or ⊃ in older books). Thus, of the four sentences 2 + 2 = 4 ⇒ 2 + 3 = 5 6 2 + 2 = 4 ⇒ 2 + 3 = 7 2 + 2 = 6 ⇒ 2 + 3 = 5 2 + 2 = 6 ⇒ 2 + 3 = 7 the second is false and the first, third and fourth are true. This usage is in sharp contrast to the usage in everyday language. In common discourse a sentence of form if p then q or p implies q suggests a kind of causality that is that q “follows” from p. Consider for example the sentence “If Columbus discovered America, then Aristotle was a Greek.” Since Aristotle was indeed a Greek this sentence either has form If true then true or If false then true and is thus true according to the meaning of “implies” we have adopted. However, common usage would judge this sentence either false or nonsensical because there is no causal relation between Columbus’s voyage and Aristotle’s nationality. To distinguish the meaning of “implies” which we have adopted (viz. that p implies q is false precisely when p is true and q is false) from other possible meanings logicians call it material implication. This is the only meaning used in mathematics. Note that material implication is extensional in the sense that the truth value of p materially implies q depends only on the truth values of p and q and not on subtler aspects of their meanings. EQUIVALENCE. The forms ‘p if and only if q’, ‘p is equivalent to q’; and ‘p exactly when q’ all have the same meaning for the mathematician: they are true when p and q have the same truth value and false in the contrary case. Some authors use “iff” is an abbreviation for “if and only if”. The mathematical symbol for if and only if is ⇔ ( ≡ in older books). Thus of the four sentences 2 + 2 = 4 ⇔ 2 + 3 = 5 2 + 2 = 4 ⇔ 2 + 3 = 7 2 + 2 = 6 ⇔ 2 + 3 = 5 2 + 2 = 6 ⇔ 2 + 3 = 7 the first and last are true while the other two are false. 7 Evidently, p if and only if q has.
Load more

Recommended publications
  • A Note on Indicator-Functions
    proceedings of the american mathematical society Volume 39, Number 1, June 1973 A NOTE ON INDICATOR-FUNCTIONS J. MYHILL Abstract. A system has the existence-property for abstracts (existence property for numbers, disjunction-property) if when- ever V(1x)A(x), M(t) for some abstract (t) (M(n) for some numeral n; if whenever VAVB, VAor hß. (3x)A(x), A,Bare closed). We show that the existence-property for numbers and the disjunc- tion property are never provable in the system itself; more strongly, the (classically) recursive functions that encode these properties are not provably recursive functions of the system. It is however possible for a system (e.g., ZF+K=L) to prove the existence- property for abstracts for itself. In [1], I presented an intuitionistic form Z of Zermelo-Frankel set- theory (without choice and with weakened regularity) and proved for it the disfunction-property (if VAvB (closed), then YA or YB), the existence- property (if \-(3x)A(x) (closed), then h4(t) for a (closed) comprehension term t) and the existence-property for numerals (if Y(3x G m)A(x) (closed), then YA(n) for a numeral n). In the appendix to [1], I enquired if these results could be proved constructively; in particular if we could find primitive recursively from the proof of AwB whether it was A or B that was provable, and likewise in the other two cases. Discussion of this question is facilitated by introducing the notion of indicator-functions in the sense of the following Definition. Let T be a consistent theory which contains Heyting arithmetic (possibly by relativization of quantifiers).
    [Show full text]
  • Soft Set Theory First Results
    An Inlum~md computers & mathematics PERGAMON Computers and Mathematics with Applications 37 (1999) 19-31 Soft Set Theory First Results D. MOLODTSOV Computing Center of the R~msian Academy of Sciences 40 Vavilova Street, Moscow 117967, Russia Abstract--The soft set theory offers a general mathematical tool for dealing with uncertain, fuzzy, not clearly defined objects. The main purpose of this paper is to introduce the basic notions of the theory of soft sets, to present the first results of the theory, and to discuss some problems of the future. (~) 1999 Elsevier Science Ltd. All rights reserved. Keywords--Soft set, Soft function, Soft game, Soft equilibrium, Soft integral, Internal regular- ization. 1. INTRODUCTION To solve complicated problems in economics, engineering, and environment, we cannot success- fully use classical methods because of various uncertainties typical for those problems. There are three theories: theory of probability, theory of fuzzy sets, and the interval mathematics which we can consider as mathematical tools for dealing with uncertainties. But all these theories have their own difficulties. Theory of probabilities can deal only with stochastically stable phenomena. Without going into mathematical details, we can say, e.g., that for a stochastically stable phenomenon there should exist a limit of the sample mean #n in a long series of trials. The sample mean #n is defined by 1 n IZn = -- ~ Xi, n i=l where x~ is equal to 1 if the phenomenon occurs in the trial, and x~ is equal to 0 if the phenomenon does not occur. To test the existence of the limit, we must perform a large number of trials.
    [Show full text]
  • Chapter 1 Logic and Set Theory
    Chapter 1 Logic and Set Theory To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. – Ian Stewart Does God play dice? The mathematics of chaos In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. That is, a proof is a logical argument, not an empir- ical one. One must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics. An unproven proposition for which there is some sort of empirical evidence is known as a conjecture. Mathematical logic is the framework upon which rigorous proofs are built. It is the study of the principles and criteria of valid inference and demonstrations. Logicians have analyzed set theory in great details, formulating a collection of axioms that affords a broad enough and strong enough foundation to mathematical reasoning. The standard form of axiomatic set theory is denoted ZFC and it consists of the Zermelo-Fraenkel (ZF) axioms combined with the axiom of choice (C). Each of the axioms included in this theory expresses a property of sets that is widely accepted by mathematicians. It is unfortunately true that careless use of set theory can lead to contradictions. Avoiding such contradictions was one of the original motivations for the axiomatization of set theory. 1 2 CHAPTER 1. LOGIC AND SET THEORY A rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic.
    [Show full text]
  • 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”.
    [Show full text]
  • What Is Fuzzy Probability Theory?
    Foundations of Physics, Vol.30,No. 10, 2000 What Is Fuzzy Probability Theory? S. Gudder1 Received March 4, 1998; revised July 6, 2000 The article begins with a discussion of sets and fuzzy sets. It is observed that iden- tifying a set with its indicator function makes it clear that a fuzzy set is a direct and natural generalization of a set. Making this identification also provides sim- plified proofs of various relationships between sets. Connectives for fuzzy sets that generalize those for sets are defined. The fundamentals of ordinary probability theory are reviewed and these ideas are used to motivate fuzzy probability theory. Observables (fuzzy random variables) and their distributions are defined. Some applications of fuzzy probability theory to quantum mechanics and computer science are briefly considered. 1. INTRODUCTION What do we mean by fuzzy probability theory? Isn't probability theory already fuzzy? That is, probability theory does not give precise answers but only probabilities. The imprecision in probability theory comes from our incomplete knowledge of the system but the random variables (measure- ments) still have precise values. For example, when we flip a coin we have only a partial knowledge about the physical structure of the coin and the initial conditions of the flip. If our knowledge about the coin were com- plete, we could predict exactly whether the coin lands heads or tails. However, we still assume that after the coin lands, we can tell precisely whether it is heads or tails. In fuzzy probability theory, we also have an imprecision in our measurements, and random variables must be replaced by fuzzy random variables and events by fuzzy events.
    [Show full text]
  • Be Measurable Spaces. a Func- 1 1 Tion F : E G Is Measurable If F − (A) E Whenever a G
    2. Measurable functions and random variables 2.1. Measurable functions. Let (E; E) and (G; G) be measurable spaces. A func- 1 1 tion f : E G is measurable if f − (A) E whenever A G. Here f − (A) denotes the inverse!image of A by f 2 2 1 f − (A) = x E : f(x) A : f 2 2 g Usually G = R or G = [ ; ], in which case G is always taken to be the Borel σ-algebra. If E is a topological−∞ 1space and E = B(E), then a measurable function on E is called a Borel function. For any function f : E G, the inverse image preserves set operations ! 1 1 1 1 f − A = f − (A ); f − (G A) = E f − (A): i i n n i ! i [ [ 1 1 Therefore, the set f − (A) : A G is a σ-algebra on E and A G : f − (A) E is f 2 g 1 f ⊆ 2 g a σ-algebra on G. In particular, if G = σ(A) and f − (A) E whenever A A, then 1 2 2 A : f − (A) E is a σ-algebra containing A and hence G, so f is measurable. In the casef G = R, 2the gBorel σ-algebra is generated by intervals of the form ( ; y]; y R, so, to show that f : E R is Borel measurable, it suffices to show −∞that x 2E : f(x) y E for all y. ! f 2 ≤ g 2 1 If E is any topological space and f : E R is continuous, then f − (U) is open in E and hence measurable, whenever U is op!en in R; the open sets U generate B, so any continuous function is measurable.
    [Show full text]
  • Expectation and Functions of Random Variables
    POL 571: Expectation and Functions of Random Variables Kosuke Imai Department of Politics, Princeton University March 10, 2006 1 Expectation and Independence To gain further insights about the behavior of random variables, we first consider their expectation, which is also called mean value or expected value. The definition of expectation follows our intuition. Definition 1 Let X be a random variable and g be any function. 1. If X is discrete, then the expectation of g(X) is defined as, then X E[g(X)] = g(x)f(x), x∈X where f is the probability mass function of X and X is the support of X. 2. If X is continuous, then the expectation of g(X) is defined as, Z ∞ E[g(X)] = g(x)f(x) dx, −∞ where f is the probability density function of X. If E(X) = −∞ or E(X) = ∞ (i.e., E(|X|) = ∞), then we say the expectation E(X) does not exist. One sometimes write EX to emphasize that the expectation is taken with respect to a particular random variable X. For a continuous random variable, the expectation is sometimes written as, Z x E[g(X)] = g(x) d F (x). −∞ where F (x) is the distribution function of X. The expectation operator has inherits its properties from those of summation and integral. In particular, the following theorem shows that expectation preserves the inequality and is a linear operator. Theorem 1 (Expectation) Let X and Y be random variables with finite expectations. 1. If g(x) ≥ h(x) for all x ∈ R, then E[g(X)] ≥ E[h(X)].
    [Show full text]
  • More on Discrete Random Variables and Their Expectations
    MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2018 Lecture 6 MORE ON DISCRETE RANDOM VARIABLES AND THEIR EXPECTATIONS Contents 1. Comments on expected values 2. Expected values of some common random variables 3. Covariance and correlation 4. Indicator variables and the inclusion-exclusion formula 5. Conditional expectations 1COMMENTSON EXPECTED VALUES (a) Recall that E is well defined unless both sums and [X] x:x<0 xpX (x) E x:x>0 xpX (x) are infinite. Furthermore, [X] is well-defined and finite if and only if both sums are finite. This is the same as requiring! that ! E[|X|]= |x|pX (x) < ∞. x " Random variables that satisfy this condition are called integrable. (b) Noter that fo any random variable X, E[X2] is always well-defined (whether 2 finite or infinite), because all the terms in the sum x x pX (x) are nonneg- ative. If we have E[X2] < ∞,we say that X is square integrable. ! (c) Using the inequality |x| ≤ 1+x 2 ,wehave E [|X|] ≤ 1+E[X2],which shows that a square integrable random variable is always integrable. Simi- larly, for every positive integer r,ifE [|X|r] is finite then it is also finite for every l<r(fill details). 1 Exercise 1. Recall that the r-the central moment of a random variable X is E[(X − E[X])r].Showthat if the r -th central moment of an almost surely non-negative random variable X is finite, then its l-th central moment is also finite for every l<r. (d) Because of the formula var(X)=E[X2] − (E[X])2,wesee that: (i) if X is square integrable, the variance is finite; (ii) if X is integrable, but not square integrable, the variance is infinite; (iii) if X is not integrable, the variance is undefined.
    [Show full text]
  • Juhani Pallasmaa, Architect, Professor Emeritus
    1 CROSSING WORLDS: MATHEMATICAL LOGIC, PHILOSOPHY, ART Honouring Juliette Kennedy University oF Helsinki, Small Hall Friday, 3 June, 2016-05-28 Juhani Pallasmaa, Architect, ProFessor Emeritus Draft 28 May 2016 THE SIXTH SENSE - diffuse perception, mood and embodied Wisdom ”Whether people are fully conscious oF this or not, they actually derive countenance and sustenance From the atmosphere oF things they live in and with”.1 Frank Lloyd Wright . --- Why do certain spaces and places make us Feel a strong aFFinity and emotional identification, while others leave us cold, or even frighten us? Why do we feel as insiders and participants in some spaces, Whereas others make us experience alienation and ”existential outsideness”, to use a notion of Edward Relph ?2 Isn’t it because the settings of the first type embrace and stimulate us, make us willingly surrender ourselves to them, and feel protected and sensually nourished? These spaces, places and environments strengthen our sense of reality and selF, whereas disturbing and alienating settings weaken our sense oF identity and reality. Resonance with the cosmos and a distinct harmonious tuning were essential qualities oF architecture since the Antiquity until the instrumentalized and aestheticized construction of the industrial era. Historically, the Fundamental task oF architecture Was to create a harmonic resonance betWeen the microcosm oF the human realm and the macrocosm oF the Universe. This harmony Was sought through proportionality based on small natural numbers FolloWing Pythagorean harmonics, on Which the harmony oF the Universe Was understood to be based. The Renaissance era also introduced the competing proportional ideal of the Golden Section.
    [Show full text]
  • Mathematical Logic
    Copyright c 1998–2005 by Stephen G. Simpson Mathematical Logic Stephen G. Simpson December 15, 2005 Department of Mathematics The Pennsylvania State University University Park, State College PA 16802 http://www.math.psu.edu/simpson/ This is a set of lecture notes for introductory courses in mathematical logic offered at the Pennsylvania State University. Contents Contents 1 1 Propositional Calculus 3 1.1 Formulas ............................... 3 1.2 Assignments and Satisfiability . 6 1.3 LogicalEquivalence. 10 1.4 TheTableauMethod......................... 12 1.5 TheCompletenessTheorem . 18 1.6 TreesandK¨onig’sLemma . 20 1.7 TheCompactnessTheorem . 21 1.8 CombinatorialApplications . 22 2 Predicate Calculus 24 2.1 FormulasandSentences . 24 2.2 StructuresandSatisfiability . 26 2.3 TheTableauMethod......................... 31 2.4 LogicalEquivalence. 37 2.5 TheCompletenessTheorem . 40 2.6 TheCompactnessTheorem . 46 2.7 SatisfiabilityinaDomain . 47 3 Proof Systems for Predicate Calculus 50 3.1 IntroductiontoProofSystems. 50 3.2 TheCompanionTheorem . 51 3.3 Hilbert-StyleProofSystems . 56 3.4 Gentzen-StyleProofSystems . 61 3.5 TheInterpolationTheorem . 66 4 Extensions of Predicate Calculus 71 4.1 PredicateCalculuswithIdentity . 71 4.2 TheSpectrumProblem . .. .. .. .. .. .. .. .. .. 75 4.3 PredicateCalculusWithOperations . 78 4.4 Predicate Calculus with Identity and Operations . ... 82 4.5 Many-SortedPredicateCalculus . 84 1 5 Theories, Models, Definability 87 5.1 TheoriesandModels ......................... 87 5.2 MathematicalTheories. 89 5.3 DefinabilityoveraModel . 97 5.4 DefinitionalExtensionsofTheories . 100 5.5 FoundationalTheories . 103 5.6 AxiomaticSetTheory . 106 5.7 Interpretability . 111 5.8 Beth’sDefinabilityTheorem. 112 6 Arithmetization of Predicate Calculus 114 6.1 Primitive Recursive Arithmetic . 114 6.2 Interpretability of PRA in Z1 ....................114 6.3 G¨odelNumbers ............................ 114 6.4 UndefinabilityofTruth. 117 6.5 TheProvabilityPredicate .
    [Show full text]
  • Mathematical Logic and Foundations of Mathematics
    Mathematical Logic and Foundations of Mathematics Stephen G. Simpson Pennsylvania State University Open House / Graduate Conference April 2–3, 2010 1 Foundations of mathematics (f.o.m.) is the study of the most basic concepts and logical structure of mathematics as a whole. Among the most basic mathematical concepts are: number, shape, set, function, algorithm, mathematical proof, mathematical definition, mathematical axiom, mathematical theorem. Some typical questions in f.o.m. are: 1. What is a number? 2. What is a shape? . 6. What is a mathematical proof? . 10. What are the appropriate axioms for mathematics? Mathematical logic gives some mathematically rigorous answers to some of these questions. 2 The concepts of “mathematical theorem” and “mathematical proof” are greatly clarified by the predicate calculus. Actually, the predicate calculus applies to non-mathematical subjects as well. Let Lxy be a 2-place predicate meaning “x loves y”. We can express properties of loving as sentences of the predicate calculus. ∀x ∃y Lxy ∃x ∀y Lyx ∀x (Lxx ⇒¬∃y Lyx) ∀x ∀y ∀z (((¬ Lyx) ∧ (¬ Lzy)) ⇒ Lzx) ∀x ((∃y Lxy) ⇒ Lxx) 3 There is a deterministic algorithm (the Tableau Method) which shows us (after a finite number of steps) that particular sentences are logically valid. For instance, the tableau ∃x (Sx ∧∀y (Eyx ⇔ (Sy ∧ ¬ Eyy))) Sa ∧∀y (Eya ⇔ Sy ∧ ¬ Eyy) Sa ∀y (Eya ⇔ Sy ∧ ¬ Eyy) Eaa ⇔ (Sa ∧ ¬ Eaa) / \ Eaa ¬ Eaa Sa ∧ ¬ Eaa ¬ (Sa ∧ ¬ Eaa) Sa / \ ¬ Sa ¬ ¬ Eaa ¬ Eaa Eaa tells us that the sentence ¬∃x (Sx ∧∀y (Eyx ⇔ (Sy ∧ ¬ Eyy))) is logically valid. This is the Russell Paradox. 4 Two significant results in mathematical logic: (G¨odel, Tarski, .
    [Show full text]
  • A Mathematical Theory of Computation?
    A Mathematical Theory of Computation? Simone Martini Dipartimento di Informatica { Scienza e Ingegneria Alma mater studiorum • Universit`adi Bologna and INRIA FoCUS { Sophia / Bologna Lille, February 1, 2017 1 / 57 Reflect and trace the interaction of mathematical logic and programming (languages), identifying some of the driving forces of this process. Previous episodes: Types HaPOC 2015, Pisa: from 1955 to 1970 (circa) Cie 2016, Paris: from 1965 to 1975 (circa) 2 / 57 Why types? Modern programming languages: control flow specification: small fraction abstraction mechanisms to model application domains. • Types are a crucial building block of these abstractions • And they are a mathematical logic concept, aren't they? 3 / 57 Why types? Modern programming languages: control flow specification: small fraction abstraction mechanisms to model application domains. • Types are a crucial building block of these abstractions • And they are a mathematical logic concept, aren't they? 4 / 57 We today conflate: Types as an implementation (representation) issue Types as an abstraction mechanism Types as a classification mechanism (from mathematical logic) 5 / 57 The quest for a \Mathematical Theory of Computation" How does mathematical logic fit into this theory? And for what purposes? 6 / 57 The quest for a \Mathematical Theory of Computation" How does mathematical logic fit into this theory? And for what purposes? 7 / 57 Prehistory 1947 8 / 57 Goldstine and von Neumann [. ] coding [. ] has to be viewed as a logical problem and one that represents a new branch of formal logics. Hermann Goldstine and John von Neumann Planning and Coding of problems for an Electronic Computing Instrument Report on the mathematical and logical aspects of an electronic computing instrument, Part II, Volume 1-3, April 1947.
    [Show full text]