DOCSLIB.ORG

First Order Logic and Nonstandard Analysis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
First Order Logic and Nonstandard Analysis First Order Logic and Nonstandard Analysis Julian Hartman September 4, 2010 Abstract This paper is intended as an exploration of nonstandard analysis, and the rigorous use of infinitesimals and infinite elements to explore properties of the real numbers. I first define and explore first order logic, and model theory. Then, I prove the compact- ness theorem, and use this to form a nonstandard structure of the real numbers. Using this nonstandard structure, it it easy to to various proofs without the use of limits that would otherwise require their use. Contents 1 Introduction 2 2 An Introduction to First Order Logic 2 2.1 Propositional Logic . 2 2.2 Logical Symbols . 2 2.3 Predicates, Constants and Functions . 2 2.4 Well-Formed Formulas . 3 3 Models 3 3.1 Structure . 3 3.2 Truth . 4 3.2.1 Satisfaction . 5 4 The Compactness Theorem 6 4.1 Soundness and Completeness . 6 5 Nonstandard Analysis 7 5.1 Making a Nonstandard Structure . 7 5.2 Applications of a Nonstandard Structure . 9 6 Sources 10 1 1 Introduction The founders of modern calculus had a less than perfect understanding of the nuts and bolts of what made it work. Both Newton and Leibniz used the notion of infinitesimal, without a rigorous understanding of what they were. Infinitely small real numbers that were still not zero was a hard thing for mathematicians to accept, and with the rigorous development of limits by the likes of Cauchy and Weierstrass, the discussion of infinitesimals subsided. Now, using first order logic for nonstandard analysis, it is possible to create a model of the real numbers that has the same properties as the traditional conception of the real numbers, but also has rigorously defined infinite and infinitesimal elements. 2 An Introduction to First Order Logic 2.1 Propositional Logic The most basic form of logic is propositional logic, an it serves a building block for first order logic. Propositional logic takes statements, and links them together with "and" or "or". Basic sentence symbols are regarded as well-formed formulas, and combine with each other in specific ways to make larger well-formed formulas. However, there are no quantifiers in this basic form of logic. First order logic remedies this lack. 2.2 Logical Symbols Predicate logic serves as a refinement of basic propositional calculus, or sentential logic. First order logic allows for the expression of more complicated conditions such as "for all" or "for some"; it allows for the quantification of the basic propositions first seen in propositional logic. There are a few basic symbols used in first order logical systems that come from propositional logic: the implication arrow ()), the negation symbol (:), the 'and' symbol (^), the inclusive 'or' symbol (_), and the inference symbol (`). In addition to these symbols from propositional logic, we introduce the quantifier symbols 8, the universal quantifier, and 9, the existential quantifier. 2.3 Predicates, Constants and Functions In first order logic, the equivalent of an english noun or noun phrase is what is called a term. Definition 1. A term is as follows: 1. A variable, or individual constant is a term. n 2. If t1; t2; :::; tn are all terms, fj (t1; t2; :::; tn) is a term, where f is a function letter. 3. An expression is a term if an only if it is a term as defined in 1 or 2. It is easiest to think of terms in the intuitive sense mentioned above as nouns that are then combined with predicate letters to form atomic formulas. Predicates define relations between different terms. 2 n Definition 2. Pj (t1; t2; :::; tn) is an atomic formula, when P is a predicate letter, and ti is a term 8 i. These atomic phrases further combine to form well-formed formulas, which are the es- sential building blocks for a first order logic. An example of an atomic formula is < xy, or x is less than y, where x and y are both terms. The symbol < is a two-place predicate symbol, which means that it defines a relation between the two components x and y. 2.4 Well-Formed Formulas Well-formed formulas, or ’wffs’, are expressions made from atomic formulas, and various quantifiers. Definition 3. The classification of expressions as a Well-Formed Formula is dependent on three rules: 1. Every atomic formula is a wff. 2. If A and B are wfs, and x is a variable, then (:A), A)B, and ((8y)A) are wffs. 3. An expression is a wff if an only if it is a wff as defined in 1 or 2. Though by the definition of a well-formed formula, the above example of an atomic for- mula also serves as an example of a wff, you can make more complicated wffs out of the atomic phrases. For example, the phrase < xy ) : ≈ xy is a wff because "x is less than y" and "x equals y" are both atomic formulas, and the implication and negation are allowed under section two of the definition. Note: At first, it would seem that the quantifier 9 cannot be in a wff, due to the defini- tion's lack of provision for this quantifier. However, upon further examination we see that 9 is logically equivalent to :((8y):X). That is, "there exists y such that y has property X" is logically equivalent to the statement "it is not true that for all y, y does not have the property X". Therefore, wffs can use the existential quantifier. Using well formed phrases and quantifier logic, it becomes much easier to translate various phrases from english to mathematical language that it was using propositional logic. For example, the sentence, "Every person has a mother." can be translated as "8y, 9x such that xMy, with aMb indicating that a is the mother of b. 3 Models 3.1 Structure A structure of a first order logical language is roughly defined as the universe in which a first order logic operates. All of the parameters of a first order language must be accounted for in the structure. 3 Definition 4. A structure, Φ, of a first order language consists of a set and functions assigning the set of parameters of the first order language to specific aspects of the structure's set. 1. The structure, Φ assigns the universal quantifier symbol (8) some nonempty set, φ. n 2. For each n-place predicate symbol with n different components, Pj , Φ assigns an n-place relation in φ n n Φ 3. For each n-place function with n inputs, fj , Φ assigns an n-place operation, (fj ) , from φn to φ. 4. For each constant c, Φ assigns a constant cΦ in the universe φ For an example of a structure, let us consider a language with only the 2-place predicate relation aDb meaning a is more delicious than b. Now, we want to assign a structure to this language. If we use a structure with the universe, φ being all the different types of fruit, and DΦ being the set of pairs of fruit such that one fruit is more delicious than the other. This satisfies parts one and two of the definition of structure, and as this particular language has no functions, part three is also satisfied. For an example of the structure of a language with an operation, let us turn to the lan- guage of ordered abelian groups. Ordered abelian groups have one 2-place predicate relation, ≤, where a ≤ b means that a is less than or equal to b. An ordered abelian group has several axioms that it must satisfy, including having a binary operation, and inverses. The operation must be associative, and communative. The group must be closed, and there must be an identity element, such that the binary operation between this element and any other element does not change the latter element. A structure which satisfies this language is the integers, with the operation addition. The ≤ relation is preserved in the integers, and addition a the binary, communicative, associative operation as specified. Negative and positive numbers are inverses of each-other, and 0 is the identity element. The integers are also closed under addition. In the language of well-formed formulas, the integers, variables, and expressions such as x + y, or −y, or z + 5 are considered terms. Expressions such as 5 ≤ 4 or x + y ≤ −6 are atomic phrases, the most basic well-formulated formulas. These can be combined into more complex wffs, such as 8x , x + 5 ≤ x + 10 3.2 Truth A well-formulated formula is not always true, it is just generated in a specific way outlined above. For example, in the model and structure of abelian groups outlined above, the statement 3 + 5 ≤ 1 is a wff, although it is clearly not true. Similarly, the statement: there exists an integer, x, such that for every integer y, x is less than or equal to y, is a wff, but is not true. 4 If a sentence is true in a structure Φ, we say that that structure is a model of that sentence. Extending this, a structure Φ is a model for a set of wffs if and only if every wff in the set is true for Φ. However, to make a statement about truth, it is necessary to give a rigorous definition of truth. To this end, we introduce the concept of satisfaction. 3.2.1 Satisfaction Let Φ be a structure of a language β. φ is the domain, or universe, of Φ. Σ is the set of all the sequences in φ, the domain of the structure Φ.
Load more

Recommended publications
  • Verifying the Unification Algorithm In
    Verifying the Uni¯cation Algorithm in LCF Lawrence C. Paulson Computer Laboratory Corn Exchange Street Cambridge CB2 3QG England August 1984 Abstract. Manna and Waldinger's theory of substitutions and uni¯ca- tion has been veri¯ed using the Cambridge LCF theorem prover. A proof of the monotonicity of substitution is presented in detail, as an exam- ple of interaction with LCF. Translating the theory into LCF's domain- theoretic logic is largely straightforward. Well-founded induction on a complex ordering is translated into nested structural inductions. Cor- rectness of uni¯cation is expressed using predicates for such properties as idempotence and most-generality. The veri¯cation is presented as a series of lemmas. The LCF proofs are compared with the original ones, and with other approaches. It appears di±cult to ¯nd a logic that is both simple and exible, especially for proving termination. Contents 1 Introduction 2 2 Overview of uni¯cation 2 3 Overview of LCF 4 3.1 The logic PPLAMBDA :::::::::::::::::::::::::: 4 3.2 The meta-language ML :::::::::::::::::::::::::: 5 3.3 Goal-directed proof :::::::::::::::::::::::::::: 6 3.4 Recursive data structures ::::::::::::::::::::::::: 7 4 Di®erences between the formal and informal theories 7 4.1 Logical framework :::::::::::::::::::::::::::: 8 4.2 Data structure for expressions :::::::::::::::::::::: 8 4.3 Sets and substitutions :::::::::::::::::::::::::: 9 4.4 The induction principle :::::::::::::::::::::::::: 10 5 Constructing theories in LCF 10 5.1 Expressions ::::::::::::::::::::::::::::::::
    [Show full text]
  • Logic- Sentence and Propositions
    Sentence and Proposition Sentence Sentence= वा啍य, Proposition= त셍कवा啍य/ प्रततऻप्‍तत=Statement Sentence is the unit or part of some language. Sentences can be expressed in all tense (present, past or future) Sentence is not able to explain quantity and quality. A sentence may be interrogative (प्रश्नवाच셍), simple or declarative (घोषड़ा配म셍), command or imperative (आदेशा配म셍), exclamatory (ववमयतबोध셍), indicative (तनदेशा配म셍), affirmative (भावा配म셍), negative (तनषेधा配म셍), skeptical (संदेहा配म셍) etc. Note : Only indicative sentences (सं셍ेता配म셍तवा啍य) are proposition. Philosophy Dept, E- contents-01( Logic-B.A Sem.III & IV) by Dr Rajendra Kr.Verma 1 All kind of sentences are not proposition, only those sentences are called proposition while they will determine or evaluate in terms of truthfulness or falsity. Sentences are governed by its own grammar. (for exp- sentence of hindi language govern by hindi grammar) Sentences are correct or incorrect / pure or impure. Sentences are may be either true or false. Philosophy Dept, E- contents-01( Logic-B.A Sem.III & IV) by Dr Rajendra Kr.Verma 2 Proposition Proposition are regarded as the material of our reasoning and we also say that proposition and statements are regarded as same. Proposition is the unit of logic. Proposition always comes in present tense. (sentences - all tenses) Proposition can explain quantity and quality. (sentences- cannot) Meaning of sentence is called proposition. Sometime more then one sentences can expressed only one proposition. Example : 1. ऩानीतबरसतरहातहैत.(Hindi) 2. ऩावुष तऩड़तोत (Sanskrit) 3. It is raining (English) All above sentences have only one meaning or one proposition.
    [Show full text]
  • Axiomatic Set Teory P.D.Welch
    Axiomatic Set Teory P.D.Welch. August 16, 2020 Contents Page 1 Axioms and Formal Systems 1 1.1 Introduction 1 1.2 Preliminaries: axioms and formal systems. 3 1.2.1 The formal language of ZF set theory; terms 4 1.2.2 The Zermelo-Fraenkel Axioms 7 1.3 Transfinite Recursion 9 1.4 Relativisation of terms and formulae 11 2 Initial segments of the Universe 17 2.1 Singular ordinals: cofinality 17 2.1.1 Cofinality 17 2.1.2 Normal Functions and closed and unbounded classes 19 2.1.3 Stationary Sets 22 2.2 Some further cardinal arithmetic 24 2.3 Transitive Models 25 2.4 The H sets 27 2.4.1 H - the hereditarily finite sets 28 2.4.2 H - the hereditarily countable sets 29 2.5 The Montague-Levy Reflection theorem 30 2.5.1 Absoluteness 30 2.5.2 Reflection Theorems 32 2.6 Inaccessible Cardinals 34 2.6.1 Inaccessible cardinals 35 2.6.2 A menagerie of other large cardinals 36 3 Formalising semantics within ZF 39 3.1 Definite terms and formulae 39 3.1.1 The non-finite axiomatisability of ZF 44 3.2 Formalising syntax 45 3.3 Formalising the satisfaction relation 46 3.4 Formalising definability: the function Def. 47 3.5 More on correctness and consistency 48 ii iii 3.5.1 Incompleteness and Consistency Arguments 50 4 The Constructible Hierarchy 53 4.1 The L -hierarchy 53 4.2 The Axiom of Choice in L 56 4.3 The Axiom of Constructibility 57 4.4 The Generalised Continuum Hypothesis in L.
    [Show full text]
  • Propositional Logic
    Propositional Logic - Review Predicate Logic - Review Propositional Logic: formalisation of reasoning involving propositions Predicate (First-order) Logic: formalisation of reasoning involving predicates. • Proposition: a statement that can be either true or false. • Propositional variable: variable intended to represent the most • Predicate (sometimes called parameterized proposition): primitive propositions relevant to our purposes a Boolean-valued function. • Given a set S of propositional variables, the set F of propositional formulas is defined recursively as: • Domain: the set of possible values for a predicate’s Basis: any propositional variable in S is in F arguments. Induction step: if p and q are in F, then so are ⌐p, (p /\ q), (p \/ q), (p → q) and (p ↔ q) 1 2 Predicate Logic – Review cont’ Predicate Logic - Review cont’ Given a first-order language L, the set F of predicate (first-order) •A first-order language consists of: formulas is constructed inductively as follows: - an infinite set of variables Basis: any atomic formula in L is in F - a set of predicate symbols Inductive step: if e and f are in F and x is a variable in L, - a set of constant symbols then so are the following: ⌐e, (e /\ f), (e \/ f), (e → f), (e ↔ f), ∀ x e, ∃ s e. •A term is a variable or a constant symbol • An occurrence of a variable x is free in a formula f if and only •An atomic formula is an expression of the form p(t1,…,tn), if it does not occur within a subformula e of f of the form ∀ x e where p is a n-ary predicate symbol and each ti is a term.
    [Show full text]
  • Unification: a Multidisciplinary Survey
    Unification: A Multidisciplinary Survey KEVIN KNIGHT Computer Science Department, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213-3890 The unification problem and several variants are presented. Various algorithms and data structures are discussed. Research on unification arising in several areas of computer science is surveyed, these areas include theorem proving, logic programming, and natural language processing. Sections of the paper include examples that highlight particular uses of unification and the special problems encountered. Other topics covered are resolution, higher order logic, the occur check, infinite terms, feature structures, equational theories, inheritance, parallel algorithms, generalization, lattices, and other applications of unification. The paper is intended for readers with a general computer science background-no specific knowledge of any of the above topics is assumed. Categories and Subject Descriptors: E.l [Data Structures]: Graphs; F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems- Computations on discrete structures, Pattern matching; Ll.3 [Algebraic Manipulation]: Languages and Systems; 1.2.3 [Artificial Intelligence]: Deduction and Theorem Proving; 1.2.7 [Artificial Intelligence]: Natural Language Processing General Terms: Algorithms Additional Key Words and Phrases: Artificial intelligence, computational complexity, equational theories, feature structures, generalization, higher order logic, inheritance, lattices, logic programming, natural language
    [Show full text]
  • CHAPTER 14 Hilbert System for Predicate Logic 1 Completeness
    CHAPTER 14 Hilbert System for Predicate Logic 1 Completeness Theorem for First Order Logic There are many proofs of the Completeness Theorem for First Order Logic. We follow here a version of Henkin's proof, as presented in the Handbook of Mathe- matical Logic. It contains a method for reducing certain problems of first-order logic back to problems about propositional logic. We give independent proof of Compactness Theorem for propositional logic. The Compactness Theorem for first-order logic and L¨owenheim-Skolem Theorems and the G¨odelCompleteness Theorem fall out of the Henkin method. 1.1 Compactness Theorem for Propositional Logic Let L = L(P; F; C) be a first order language with equality. We assume that the sets P, F, C are infinitely enumerable. We define a propositional logic within it as follows. Prime formulas We consider a subset P of the set F of all formulas of L. Intuitively these are formulas of L which are not direct propositional com- bination of simpler formulas, that is, atomic formulas (AF) and formulas beginning with quantifiers. Formally, we have that P = fA 2 F : A 2 AF or A = 8xB; A = 9xB for B 2 Fg: Example 1.1 The following are primitive formulas. R(t1; t2); 8x(A(x) ):A(x)); (c = c); 9x(Q(x; y) \ 8yA(y)). The following are not primitive formulas. (R(t1; t2) ) (c = c)); (R(t1; t2) [ 8x(A(x) ):A(x)). Given a set P of primitive formulas we define in a standard way the set P F of propositional formulas as follows.
    [Show full text]
  • Self-Similarity in the Foundations
    Self-similarity in the Foundations Paul K. Gorbow Thesis submitted for the degree of Ph.D. in Logic, defended on June 14, 2018. Supervisors: Ali Enayat (primary) Peter LeFanu Lumsdaine (secondary) Zachiri McKenzie (secondary) University of Gothenburg Department of Philosophy, Linguistics, and Theory of Science Box 200, 405 30 GOTEBORG,¨ Sweden arXiv:1806.11310v1 [math.LO] 29 Jun 2018 2 Contents 1 Introduction 5 1.1 Introductiontoageneralaudience . ..... 5 1.2 Introduction for logicians . .. 7 2 Tour of the theories considered 11 2.1 PowerKripke-Plateksettheory . .... 11 2.2 Stratifiedsettheory ................................ .. 13 2.3 Categorical semantics and algebraic set theory . ....... 17 3 Motivation 19 3.1 Motivation behind research on embeddings between models of set theory. 19 3.2 Motivation behind stratified algebraic set theory . ...... 20 4 Logic, set theory and non-standard models 23 4.1 Basiclogicandmodeltheory ............................ 23 4.2 Ordertheoryandcategorytheory. ...... 26 4.3 PowerKripke-Plateksettheory . .... 28 4.4 First-order logic and partial satisfaction relations internal to KPP ........ 32 4.5 Zermelo-Fraenkel set theory and G¨odel-Bernays class theory............ 36 4.6 Non-standardmodelsofsettheory . ..... 38 5 Embeddings between models of set theory 47 5.1 Iterated ultrapowers with special self-embeddings . ......... 47 5.2 Embeddingsbetweenmodelsofsettheory . ..... 57 5.3 Characterizations.................................. .. 66 6 Stratified set theory and categorical semantics 73 6.1 Stratifiedsettheoryandclasstheory . ...... 73 6.2 Categoricalsemantics ............................... .. 77 7 Stratified algebraic set theory 85 7.1 Stratifiedcategoriesofclasses . ..... 85 7.2 Interpretation of the Set-theories in the Cat-theories ................ 90 7.3 ThesubtoposofstronglyCantorianobjects . ....... 99 8 Where to go from here? 103 8.1 Category theoretic approach to embeddings between models of settheory .
    [Show full text]
  • Monadic Decomposition
    Monadic Decomposition Margus Veanes1, Nikolaj Bjørner1, Lev Nachmanson1, and Sergey Bereg2 1 Microsoft Research {margus,nbjorner,levnach}@microsoft.com 2 The University of Texas at Dallas [email protected] Abstract. Monadic predicates play a prominent role in many decid- able cases, including decision procedures for symbolic automata. We are here interested in discovering whether a formula can be rewritten into a Boolean combination of monadic predicates. Our setting is quantifier- free formulas over a decidable background theory, such as arithmetic and we here develop a semi-decision procedure for extracting a monadic decomposition of a formula when it exists. 1 Introduction Classical decidability results of fragments of logic [7] are based on careful sys- tematic study of restricted cases either by limiting allowed symbols of the lan- guage, limiting the syntax of the formulas, fixing the background theory, or by using combinations of such restrictions. Many decidable classes of problems, such as monadic first-order logic or the L¨owenheim class [29], the L¨ob-Gurevich class [28], monadic second-order logic with one successor (S1S) [8], and monadic second-order logic with two successors (S2S) [35] impose at some level restric- tions to monadic or unary predicates to achieve decidability. Here we propose and study an orthogonal problem of whether and how we can transform a formula that uses multiple free variables into a simpler equivalent formula, but where the formula is not a priori syntactically or semantically restricted to any fixed fragment of logic. Simpler in this context means that we have eliminated all theory specific dependencies between the variables and have transformed the formula into an equivalent Boolean combination of predicates that are “essentially” unary.
    [Show full text]
  • Foundations of Mathematics
    Foundations of Mathematics November 27, 2017 ii Contents 1 Introduction 1 2 Foundations of Geometry 3 2.1 Introduction . .3 2.2 Axioms of plane geometry . .3 2.3 Non-Euclidean models . .4 2.4 Finite geometries . .5 2.5 Exercises . .6 3 Propositional Logic 9 3.1 The basic definitions . .9 3.2 Disjunctive Normal Form Theorem . 11 3.3 Proofs . 12 3.4 The Soundness Theorem . 19 3.5 The Completeness Theorem . 20 3.6 Completeness, Consistency and Independence . 22 4 Predicate Logic 25 4.1 The Language of Predicate Logic . 25 4.2 Models and Interpretations . 27 4.3 The Deductive Calculus . 29 4.4 Soundness Theorem for Predicate Logic . 33 5 Models for Predicate Logic 37 5.1 Models . 37 5.2 The Completeness Theorem for Predicate Logic . 37 5.3 Consequences of the completeness theorem . 41 5.4 Isomorphism and elementary equivalence . 43 5.5 Axioms and Theories . 47 5.6 Exercises . 48 6 Computability Theory 51 6.1 Introduction and Examples . 51 6.2 Finite State Automata . 52 6.3 Exercises . 55 6.4 Turing Machines . 56 6.5 Recursive Functions . 61 6.6 Exercises . 67 iii iv CONTENTS 7 Decidable and Undecidable Theories 69 7.1 Introduction . 69 7.1.1 Gödel numbering . 69 7.2 Decidable vs. Undecidable Logical Systems . 70 7.3 Decidable Theories . 71 7.4 Gödel’s Incompleteness Theorems . 74 7.5 Exercises . 81 8 Computable Mathematics 83 8.1 Computable Combinatorics . 83 8.2 Computable Analysis . 85 8.2.1 Computable Real Numbers . 85 8.2.2 Computable Real Functions .
    [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]
  • Shelah's Singular Compactness Theorem 0
    SHELAH’S SINGULAR COMPACTNESS THEOREM PAUL C. EKLOF Abstract. We present Shelah’s famous theorem in a version for modules, together with a self-contained proof and some ex- amples. This exposition is based on lectures given at CRM in October 2006. 0. Introduction The Singular Compactness Theorem is about an abstract notion of “free”. The general form of the theorem is as follows: If λ is a singular cardinal and M is a λ-generated module such that enough < κ-generated submodules are “free” for sufficiently many regular κ < λ, then M is “free”. Of course, for this to have any chance to be a theorem (of ZFC) there need to be assumptions on the notion of “free”. These are detailed in the next section along with a precise statement of the theorem (Theorem 1.4), including a precise definition of “enough”. Another version of the Singular Compactness Theorem—with a dif- ferent definition of “enough”—is given at the start of section 3. The proof of Theorem 1.4 is given in sections 3 and 4; some examples and applications are given in sections 2 and 5. First a few words about the history of the theorem. 0.1 History. In 1973 Saharon Shelah proved that the Whitehead problem for abelian groups of cardinality ℵ1 is undecidable in ZFC; in particular he showed under the assumption of the Axiom of Con- structibility, V = L, that all Whitehead groups of cardinality ℵ1 are free (see [13]). His argument easily extended, by induction, to prove that for all n ∈ ω, Whitehead groups of cardinality ℵn are free.
    [Show full text]
  • 3. Temporal Logics and Model Checking
    3. Temporal Logics and Model Checking Page Temporal Logics 3.2 Linear Temporal Logic (PLTL) 3.4 Branching Time Temporal Logic (BTTL) 3.8 Computation Tree Logic (CTL) 3.9 Linear vs. Branching Time TL 3.16 Structure of Model Checker 3.19 Notion of Fixpoint 3.20 Fixpoint Characterization of CTL 3.25 CTL Model Checking Algorithm 3.30 Symbolic Model Checking 3.34 Model Checking Tools 3.42 References 3.46 3.1 (of 47) Temporal Logics Temporal Logics • Temporal logic is a type of modal logic that was originally developed by philosophers to study different modes of “truth” • Temporal logic provides a formal system for qualitatively describing and reasoning about how the truth values of assertions change over time • It is appropriate for describing the time-varying behavior of systems (or programs) Classification of Temporal Logics • The underlying nature of time: Linear: at any time there is only one possible future moment, linear behavioral trace Branching: at any time, there are different possible futures, tree-like trace structure • Other considerations: Propositional vs. first-order Point vs. intervals Discrete vs. continuous time Past vs. future 3.2 (of 47) Linear Temporal Logic • Time lines Underlying structure of time is a totally ordered set (S,<), isomorphic to (N,<): Discrete, an initial moment without predecessors, infinite into the future. • Let AP be set of atomic propositions, a linear time structure M=(S, x, L) S: a set of states x: NS an infinite sequence of states, (x=s0,s1,...) L: S2AP labeling each state with the set of atomic propositions in AP true at the state.
    [Show full text]