DOCSLIB.ORG

MATHEMATICAL LOGIC Narrowly Construed, Mathematical Logic Is the Study of Definition and Inference in YIANNIS N

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

MATHEMATICAL LOGIC Narrowly construed, mathematical logic is the study of definition and inference in YIANNIS N. MOSCHOVAKIS mathematical models of fragments of lan- guage, especially the first order logic frag- ment. Logic has made critical contributions I. Propositional Logic, PL . to the foundations of science, especially L II. First Order Logic, FO . through the work of Kurt G¨odel, and it III. G¨odel’s Incompleteness Theorem. also has numerous applications. For set the- IV. Computability. ory theoretical computer science V. Recursion and Programming. and , these VI. Alternative Logics. applications are so important, that parts VII. Set Theory. of these fields are normally included in the modern, broad conception of the discipline. Glossary I. Propositional Logic, PL Church-Turing Thesis: Claim that ev- ery computable function can be computed Each logic L has a syntax which delin- by a Turing machine. eates the grammatically correct linguistic Computability theory: Study of com- expressions of L, a semantics which assigns putable functions on the natural numbers. meaning to the correct expressions, and a Continuum hypothesis: Conjecture structured system of proofs which specifies that there are only two sizes of infinite sets the rules by which some L-expressions can of real numbers. be inferred from others. Database: Finite, typically relational There are other words to describe these structure. things: formal language is sometimes used First order logic: Mathematical model to describe a plain syntax, formal system of the part of language built up from the pro- often identifies a syntax together with an positional connectives and the quantifiers. inference system (but without an interpre- Incompleteness phenomenon: G¨odel’s tation), and abstract logic has been used to discovery, that sufficiently strong axiomatic refer to a syntax together with an interpreta- theories cannot decide all propositions which tion, leaving inference aside. It is, however, they can express. a fundamental feature of logic that it draws Model theory: Study of formal defin- clean distinctions and studies the connec- ability in first order structures. tions among these three aspects of language. Paradox: Counterintuitive truth. We explain them first in the simplest exam- Peano arithmetic: Axiomatic theory of ple of the “logic of propositions”, which is natural numbers. part of many important logics. Proof theory: Study of inference in for- A. Propositional Syntax mal systems independently of their interpre- tation. The symbols of PL are the connectives Propositional connectives: The lin- ¬ (not) & (and) ∨ (or) guistic constructs “and”, “not”, “or” and “implies”. → (implies, if-then) Quantifiers: The linguistic constructs the two parentheses ‘(’, ‘)’, and an infi- “there exists” and “for all”. nite list of (formal) propositional variables Turing machine: Mathematical model P0, P1, P2,... which intuitively stand for of computing device with unbounded mem- declarative propositions, things like ‘John ory. loves Mary’ or ‘3 is a prime number’. It has Unsolvable problem: A problem whose only one category of grammatically correct solution requires a non-existent algorithm. expressions, the formulas, which are strings 1 2 YIANNIS N. MOSCHOVAKIS (finite sequences) of symbols defined induc- what the truth value of B, so that ‘if the tively by the following conditions: moon is made of cheese, then 1 + 1 = 5’ is F1. Each Pi is a formula. true (on the plausible assumption that the F2. If A and B are formulas, then so are moon is not made of cheese). This material the expressions implication assumed by Propositional Logic has been attacked as counterintuitive, but it ¬A (A & B) (A ∨ B) (A → B) agrees with mathematical practice and it is For example, if P and Q are propositional the only useful interpretation of implication variables, then (P → Q) and (P ∨¬P ) are which accords with the Compositionality formulas, which we read as “if P then Q” Principle. and “either P or not P ”. Using these rules, we can construct for The inductive definition gives a precise each formula A a truth table which tabulates specification of exactly which strings of sym- its truth value under all assignments of truth bols are formulas, and also insures that each values to the variables. For example, the formula is either prime, i.e., just a variable truth table for (Q → P ) consists of the first Pi, or it can be constructed in exactly one three columns of Table 2 while the first two way from its simpler immediate parts, by one of the connectives. This makes it possible to P Q (Q → P ) (P → (Q → P )) prove properties of formulas and to define 1 1 1 1 operations on them by structural induction 1 0 1 1 on their definition. 0 0 1 1 More propositional connectives can be in- 0 1 0 1 troduced as “abbreviations” of formula com- Table 2. binations, e.g., and the last column give the truth table for A ↔ B ≡ ((A → B)&(B → A)) (P → (Q → P )) . A ∨ B ∨ C ≡ (A ∨ (B ∨ C)). If n variables occur in a formula A, then the truth table for A has 2n rows and de- B. Propositional Semantics termines an n-ary bit function vA, with ar- If B stands for some true proposition, guments and values in the two-element set then ¬B is false, independently of the {1, 0}. By the Definitional Completeness “meaning” or internal structure of B. This Theorem, every n-ary bit function is vA for is an instance of a general Compositional- some A, so that the formulas of PL provide ity Principle for PL: The truth value of a definitions (or “symbolic representations”) formula depends only on the truth values for all bit functions. of its immediate parts. The semantics of A formula A is a semantic consequence PL comprise the rules for computing truth of a set of formulas T (or T -valid) if every values, and they can be summarized in Ta- assignment to the variables which satisfies ble 1, where 1 stands for ‘truth’ and 0 for (makes true) all the formulas in T also sat- ‘falsity’. By the first line of this table, for isfies A,. We write A B ¬A (A & B) (A ∨ B) (A → B) T |= A ⇔ A is T -valid, 1 1 0 1 1 1 and |= A, in the important special case when 1 0 0 0 1 0 T is empty, in which case A is called a tau- 0 1 1 0 1 1 tology. A formula A is satisfiable if some 0 0 1 0 0 1 assignment satisfies it, i.e., if ¬A is not a Table 1. Truth value semantics. tautology. Let example, if A and B are both true, then ¬A A ∼ B ⇔ {A} |= B and {B} |= A is false while (A & B), (A ∨ B) and (A → B) ⇔ |= A ↔ B, are all true. Notice that if A is false, then (A → B) is reckoned to be true no matter and call A and B equivalent if A ∼ B. MATHEMATICAL LOGIC 3 Equivalent formulas define the same bit the computation of bit functions by appeal- function, and they can be substituted for ing to the formula representations of the cir- each other without changing truth values. cuits which realize them. For example, using Clearly disjunctive normal forms, one sees immedi- ately that (if we do not care about cost), (A → B) ∼ (¬A ∨ B), every n-ary bit function can be computed by so that the implication connective is super- an unbounded fan-in circuit in no more than fluous. In fact, every formula is equivalent 3 time units. There is, in general, a sub- to one in disjunctive normal form, i.e., a dis- stantial trade-off between the size and time junction A1 ∨···∨ Ak where each Ai is a complexity of the circuits which compute a conjunction of variables or negations of vari- given bit function. ables (literals). D. The Satisfiability Problem C. Applications to Circuits The assertion that “C(A) and C(B) Each formula A with n variables can be never give the same output on the same realized by a switching circuit C(A) with n inputs” means precisely that “(A ↔¬B) is inputs and one output, so that C(Pi) con- a tautology”, so that to detect that A and B sists of just one input-output edge, C(A&B) do not have this safety property we need to is constructed by joining C(A) and C(B) determine whether the formula ¬(A ↔¬B) with an and-gate, etc. Figure 1 exhibits the is satisfiable. Because of such natural formulations P 1 s of “error detection” for circuits relative & - ¬ > s - to given specifications, it is very impor- P ∨ 2 > tant to find efficient algorithms for deter- P 3 mining whether a given formula is satisfi- able. The problem is of non-deterministi- Fig. 1. The circuit for (¬((P1 & P2) ∨ P3). cally polynomial time complexity (NP), be- circuit for ((P1 &P2) → P3) using the equiv- cause it can be resolved by guessing (“non- alent formula without implications, so that deterministically”) some assignment and only ¬-, &- and ∨-gates are required. These then verifying that it satisfies A in a number are restricted circuits, of fan-in (maximum of steps which is bounded by a polynomial number of edges into a node) 2 and fan-out in the length of A; and it is NP-complete, 1, but the Definitional Completeness The- i.e., every NP-problem can be “reduced” to orem implies that every n-ary bit function it by a polynomial reduction. This is a ba- can be computed by some formula circuit sic result of S. Cook, who introduced the C(A). complexity class NP, showed that it con- There are basically two useful measures tains a large number of important problems, of circuit complexity, and both of them are and asked if it coincides with the (seem- faithfully mirrored in formulas.
Recommended publications
  • The Enigmatic Number E: a History in Verse and Its Uses in the Mathematics Classroom

    The Enigmatic Number E: a History in Verse and Its Uses in the Mathematics Classroom

    To appear in MAA Loci: Convergence The Enigmatic Number e: A History in Verse and Its Uses in the Mathematics Classroom Sarah Glaz Department of Mathematics University of Connecticut Storrs, CT 06269 [email protected] Introduction In this article we present a history of e in verse—an annotated poem: The Enigmatic Number e . The annotation consists of hyperlinks leading to biographies of the mathematicians appearing in the poem, and to explanations of the mathematical notions and ideas presented in the poem. The intention is to celebrate the history of this venerable number in verse, and to put the mathematical ideas connected with it in historical and artistic context. The poem may also be used by educators in any mathematics course in which the number e appears, and those are as varied as e's multifaceted history. The sections following the poem provide suggestions and resources for the use of the poem as a pedagogical tool in a variety of mathematics courses. They also place these suggestions in the context of other efforts made by educators in this direction by briefly outlining the uses of historical mathematical poems for teaching mathematics at high-school and college level. Historical Background The number e is a newcomer to the mathematical pantheon of numbers denoted by letters: it made several indirect appearances in the 17 th and 18 th centuries, and acquired its letter designation only in 1731. Our history of e starts with John Napier (1550-1617) who defined logarithms through a process called dynamical analogy [1]. Napier aimed to simplify multiplication (and in the same time also simplify division and exponentiation), by finding a model which transforms multiplication into addition.
  • Zero-One Laws

    Zero-One Laws

    1 THE LOGIC IN COMPUTER SCIENCE COLUMN by 2 Yuri GUREVICH Electrical Engineering and Computer Science UniversityofMichigan, Ann Arb or, MI 48109-2122, USA [email protected] Zero-One Laws Quisani: I heard you talking on nite mo del theory the other day.Itisinteresting indeed that all those famous theorems ab out rst-order logic fail in the case when only nite structures are allowed. I can understand that for those, likeyou, educated in the tradition of mathematical logic it seems very imp ortantto ndoutwhichof the classical theorems can b e rescued. But nite structures are to o imp ortant all by themselves. There's got to b e deep nite mo del theory that has nothing to do with in nite structures. Author: \Nothing to do" sounds a little extremist to me. Sometimes in nite ob jects are go o d approximations of nite ob jects. Q: I do not understand this. Usually, nite ob jects approximate in nite ones. A: It may happ en that the in nite case is cleaner and easier to deal with. For example, a long nite sum may b e replaced with a simpler integral. Returning to your question, there is indeed meaningful, inherently nite, mo del theory. One exciting issue is zero- one laws. Consider, say, undirected graphs and let be a propertyofsuch graphs. For example, may b e connectivity. What fraction of n-vertex graphs have the prop erty ? It turns out that, for many natural prop erties , this fraction converges to 0 or 1asn grows to in nity. If the fraction converges to 1, the prop erty is called almost sure.
  • Basic Concepts of Set Theory, Functions and Relations 1. Basic

    Basic Concepts of Set Theory, Functions and Relations 1. Basic

    Ling 310, adapted from UMass Ling 409, Partee lecture notes March 1, 2006 p. 1 Basic Concepts of Set Theory, Functions and Relations 1. Basic Concepts of Set Theory........................................................................................................................1 1.1. Sets and elements ...................................................................................................................................1 1.2. Specification of sets ...............................................................................................................................2 1.3. Identity and cardinality ..........................................................................................................................3 1.4. Subsets ...................................................................................................................................................4 1.5. Power sets .............................................................................................................................................4 1.6. Operations on sets: union, intersection...................................................................................................4 1.7 More operations on sets: difference, complement...................................................................................5 1.8. Set-theoretic equalities ...........................................................................................................................5 Chapter 2. Relations and Functions ..................................................................................................................6
  • On the Normality of Numbers

    On the Normality of Numbers

    ON THE NORMALITY OF NUMBERS Adrian Belshaw B. Sc., University of British Columbia, 1973 M. A., Princeton University, 1976 A THESIS SUBMITTED 'IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of Mathematics @ Adrian Belshaw 2005 SIMON FRASER UNIVERSITY Fall 2005 All rights reserved. This work may not be reproduced in whole or in part, by photocopy or other means, without the permission of the author. APPROVAL Name: Adrian Belshaw Degree: Master of Science Title of Thesis: On the Normality of Numbers Examining Committee: Dr. Ladislav Stacho Chair Dr. Peter Borwein Senior Supervisor Professor of Mathematics Simon Fraser University Dr. Stephen Choi Supervisor Assistant Professor of Mathematics Simon Fraser University Dr. Jason Bell Internal Examiner Assistant Professor of Mathematics Simon Fraser University Date Approved: December 5. 2005 SIMON FRASER ' u~~~snrllbrary DECLARATION OF PARTIAL COPYRIGHT LICENCE The author, whose copyright is declared on the title page of this work, has granted to Simon Fraser University the right to lend this thesis, project or extended essay to users of the Simon Fraser University Library, and to make partial or single copies only for such users or in response to a request from the library of any other university, or other educational institution, on its own behalf or for one of its users. The author has further granted permission to Simon Fraser University to keep or make a digital copy for use in its circulating collection, and, without changing the content, to translate the thesislproject or extended essays, if technically possible, to any medium or format for the purpose of preservation of the digital work.
  • Diophantine Approximation and Transcendental Numbers

    Diophantine Approximation and Transcendental Numbers

    Diophantine Approximation and Transcendental Numbers Connor Goldstick June 6, 2017 Contents 1 Introduction 1 2 Continued Fractions 2 3 Rational Approximations 3 4 Transcendental Numbers 6 5 Irrationality Measure 7 6 The continued fraction for e 8 6.1 e's Irrationality measure . 10 7 Conclusion 11 A The Lambert Continued Fraction 11 1 Introduction Suppose we have an irrational number, α, that we want to approximate with a rational number, p=q. This question of approximating an irrational number is the primary concern of Diophantine approximation. In other words, we want jα−p=qj < . However, this method of trying to approximate α is boring, as it is possible to get an arbitrary amount of precision by making q large. To remedy this problem, it makes sense to vary with q. The problem we are really trying to solve is finding p and q such that p 1 1 jα − j < which is equivalent to jqα − pj < (1) q q2 q Solutions to this problem have applications in both number theory and in more applied fields. For example, in signal processing many of the quickest approximation algorithms are given by solutions to Diophantine approximation problems. Diophantine approximations also give many important results like the continued fraction expansion of e. One of the most interesting aspects of Diophantine approximations are its relationship with transcendental 1 numbers (a number that cannot be expressed of the root of a polynomial with rational coefficients). One of the key characteristics of a transcendental number is that it is easy to approximate with rational numbers. This paper is separated into two categories.
  • Notes on Mathematical Logic David W. Kueker

    Notes on Mathematical Logic David W. Kueker

    Notes on Mathematical Logic David W. Kueker University of Maryland, College Park E-mail address: [email protected] URL: http://www-users.math.umd.edu/~dwk/ Contents Chapter 0. Introduction: What Is Logic? 1 Part 1. Elementary Logic 5 Chapter 1. Sentential Logic 7 0. Introduction 7 1. Sentences of Sentential Logic 8 2. Truth Assignments 11 3. Logical Consequence 13 4. Compactness 17 5. Formal Deductions 19 6. Exercises 20 20 Chapter 2. First-Order Logic 23 0. Introduction 23 1. Formulas of First Order Logic 24 2. Structures for First Order Logic 28 3. Logical Consequence and Validity 33 4. Formal Deductions 37 5. Theories and Their Models 42 6. Exercises 46 46 Chapter 3. The Completeness Theorem 49 0. Introduction 49 1. Henkin Sets and Their Models 49 2. Constructing Henkin Sets 52 3. Consequences of the Completeness Theorem 54 4. Completeness Categoricity, Quantifier Elimination 57 5. Exercises 58 58 Part 2. Model Theory 59 Chapter 4. Some Methods in Model Theory 61 0. Introduction 61 1. Realizing and Omitting Types 61 2. Elementary Extensions and Chains 66 3. The Back-and-Forth Method 69 i ii CONTENTS 4. Exercises 71 71 Chapter 5. Countable Models of Complete Theories 73 0. Introduction 73 1. Prime Models 73 2. Universal and Saturated Models 75 3. Theories with Just Finitely Many Countable Models 77 4. Exercises 79 79 Chapter 6. Further Topics in Model Theory 81 0. Introduction 81 1. Interpolation and Definability 81 2. Saturated Models 84 3. Skolem Functions and Indescernables 87 4. Some Applications 91 5.
  • Network Intrusion Detection with Xgboost and Deep Learning Algorithms: an Evaluation Study

    Network Intrusion Detection with Xgboost and Deep Learning Algorithms: an Evaluation Study

    2020 International Conference on Computational Science and Computational Intelligence (CSCI) Network Intrusion Detection with XGBoost and Deep Learning Algorithms: An Evaluation Study Amr Attia Miad Faezipour Abdelshakour Abuzneid Computer Science & Engineering Computer Science & Engineering Computer Science & Engineering University of Bridgeport, CT 06604, USA University of Bridgeport, CT 06604, USA University of Bridgeport, CT 06604, USA [email protected] [email protected] [email protected] Abstract— This paper introduces an effective Network Intrusion In the KitNET model introduced in [2], an unsupervised Detection Systems (NIDS) framework that deploys incremental technique is introduced for anomaly-based intrusion statistical damping features of the packets along with state-of- detection. Incremental statistical feature extraction of the the-art machine/deep learning algorithms to detect malicious packets is passed through ensembles of autoencoders with a patterns. A comprehensive evaluation study is conducted predefined threshold. The model calculates the Root Mean between eXtreme Gradient Boosting (XGBoost) and Artificial Neural Networks (ANN) where feature selection and/or feature Square (RMS) error to detect anomaly behavior. The higher dimensionality reduction techniques such as Principal the calculated RMS at the output, the higher probability of Component Analysis (PCA) and Linear Discriminant Analysis suspicious activity. (LDA) are also integrated into the models to decrease the system Supervised learning has achieved very decent results with complexity for achieving fast responses. Several experimental algorithms such as Random Forest, ZeroR, J48, AdaBoost, runs confirm how powerful machine/deep learning algorithms Logit Boost, and Multilayer Perceptron [3]. Machine/deep are for intrusion detection on known attacks when combined learning-based algorithms for NIDS have been extensively with the appropriate features extracted.
  • The Prime Number Theorem a PRIMES Exposition

    The Prime Number Theorem a PRIMES Exposition

    The Prime Number Theorem A PRIMES Exposition Ishita Goluguri, Toyesh Jayaswal, Andrew Lee Mentor: Chengyang Shao TABLE OF CONTENTS 1 Introduction 2 Tools from Complex Analysis 3 Entire Functions 4 Hadamard Factorization Theorem 5 Riemann Zeta Function 6 Chebyshev Functions 7 Perron Formula 8 Prime Number Theorem © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 2 Introduction • Euclid (300 BC): There are infinitely many primes • Legendre (1808): for primes less than 1,000,000: x π(x) ' log x © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 3 Progress on the Distribution of Prime Numbers • Euler: The product formula 1 X 1 Y 1 ζ(s) := = ns 1 − p−s n=1 p so (heuristically) Y 1 = log 1 1 − p−1 p • Chebyshev (1848-1850): if the ratio of π(x) and x= log x has a limit, it must be 1 • Riemann (1859): On the Number of Primes Less Than a Given Magnitude, related π(x) to the zeros of ζ(s) using complex analysis • Hadamard, de la Vallée Poussin (1896): Proved independently the prime number theorem by showing ζ(s) has no zeros of the form 1 + it, hence the celebrated prime number theorem © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 4 Tools from Complex Analysis Theorem (Maximum Principle) Let Ω be a domain, and let f be holomorphic on Ω. (A) jf(z)j cannot attain its maximum inside Ω unless f is constant. (B) The real part of f cannot attain its maximum inside Ω unless f is a constant. Theorem (Jensen’s Inequality) Suppose f is holomorphic on the whole complex plane and f(0) = 1.
  • The Development of Mathematical Logic from Russell to Tarski: 1900–1935

    The Development of Mathematical Logic from Russell to Tarski: 1900–1935

    The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu Richard Zach Calixto Badesa The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu (University of California, Berkeley) Richard Zach (University of Calgary) Calixto Badesa (Universitat de Barcelona) Final Draft—May 2004 To appear in: Leila Haaparanta, ed., The Development of Modern Logic. New York and Oxford: Oxford University Press, 2004 Contents Contents i Introduction 1 1 Itinerary I: Metatheoretical Properties of Axiomatic Systems 3 1.1 Introduction . 3 1.2 Peano’s school on the logical structure of theories . 4 1.3 Hilbert on axiomatization . 8 1.4 Completeness and categoricity in the work of Veblen and Huntington . 10 1.5 Truth in a structure . 12 2 Itinerary II: Bertrand Russell’s Mathematical Logic 15 2.1 From the Paris congress to the Principles of Mathematics 1900–1903 . 15 2.2 Russell and Poincar´e on predicativity . 19 2.3 On Denoting . 21 2.4 Russell’s ramified type theory . 22 2.5 The logic of Principia ......................... 25 2.6 Further developments . 26 3 Itinerary III: Zermelo’s Axiomatization of Set Theory and Re- lated Foundational Issues 29 3.1 The debate on the axiom of choice . 29 3.2 Zermelo’s axiomatization of set theory . 32 3.3 The discussion on the notion of “definit” . 35 3.4 Metatheoretical studies of Zermelo’s axiomatization . 38 4 Itinerary IV: The Theory of Relatives and Lowenheim’s¨ Theorem 41 4.1 Theory of relatives and model theory . 41 4.2 The logic of relatives .
  • Definitions and Nondefinability in Geometry 475 2

    Definitions and Nondefinability in Geometry 475 2

    Definitions and Nondefinability in Geometry1 James T. Smith Abstract. Around 1900 some noted mathematicians published works developing geometry from its very beginning. They wanted to supplant approaches, based on Euclid’s, which han- dled some basic concepts awkwardly and imprecisely. They would introduce precision re- quired for generalization and application to new, delicate problems in higher mathematics. Their work was controversial: they departed from tradition, criticized standards of rigor, and addressed fundamental questions in philosophy. This paper follows the problem, Which geo- metric concepts are most elementary? It describes a false start, some successful solutions, and an argument that one of those is optimal. It’s about axioms, definitions, and definability, and emphasizes contributions of Mario Pieri (1860–1913) and Alfred Tarski (1901–1983). By fol- lowing this thread of ideas and personalities to the present, the author hopes to kindle interest in a fascinating research area and an exciting era in the history of mathematics. 1. INTRODUCTION. Around 1900 several noted mathematicians published major works on a subject familiar to us from school: developing geometry from the very beginning. They wanted to supplant the established approaches, which were based on Euclid’s, but which handled awkwardly and imprecisely some concepts that Euclid did not treat fully. They would present geometry with the precision required for general- ization and applications to new, delicate problems in higher mathematics—precision beyond the norm for most elementary classes. Work in this area was controversial: these mathematicians departed from tradition, criticized previous standards of rigor, and addressed fundamental questions in logic and philosophy of mathematics.2 After establishing background, this paper tells a story about research into the ques- tion, Which geometric concepts are most elementary? It describes a false start, some successful solutions, and a demonstration that one of those is in a sense optimal.
  • An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics

    An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics

    STUDENT MATHEMATICAL LIBRARY Volume 87 An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics Matthew Katz Jan Reimann Mathematics Advanced Study Semesters 10.1090/stml/087 An Introduction to Ramsey Theory STUDENT MATHEMATICAL LIBRARY Volume 87 An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics Matthew Katz Jan Reimann Mathematics Advanced Study Semesters Editorial Board Satyan L. Devadoss John Stillwell (Chair) Rosa Orellana Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 05D10, 03-01, 03E10, 03B10, 03B25, 03D20, 03H15. Jan Reimann was partially supported by NSF Grant DMS-1201263. For additional information and updates on this book, visit www.ams.org/bookpages/stml-87 Library of Congress Cataloging-in-Publication Data Names: Katz, Matthew, 1986– author. | Reimann, Jan, 1971– author. | Pennsylvania State University. Mathematics Advanced Study Semesters. Title: An introduction to Ramsey theory: Fast functions, infinity, and metamathemat- ics / Matthew Katz, Jan Reimann. Description: Providence, Rhode Island: American Mathematical Society, [2018] | Series: Student mathematical library; 87 | “Mathematics Advanced Study Semesters.” | Includes bibliographical references and index. Identifiers: LCCN 2018024651 | ISBN 9781470442903 (alk. paper) Subjects: LCSH: Ramsey theory. | Combinatorial analysis. | AMS: Combinatorics – Extremal combinatorics – Ramsey theory. msc | Mathematical logic and foundations – Instructional exposition (textbooks, tutorial papers, etc.). msc | Mathematical
  • Model Completeness and Relative Decidability

    Model Completeness and Relative Decidability

    Model Completeness and Relative Decidability Jennifer Chubb, Russell Miller,∗ & Reed Solomon March 5, 2019 Abstract We study the implications of model completeness of a theory for the effectiveness of presentations of models of that theory. It is im- mediate that for a computable model A of a computably enumerable, model complete theory, the entire elementary diagram E(A) must be decidable. We prove that indeed a c.e. theory T is model complete if and only if there is a uniform procedure that succeeds in deciding E(A) from the atomic diagram ∆(A) for all countable models A of T . Moreover, if every presentation of a single isomorphism type A has this property of relative decidability, then there must be a procedure with succeeds uniformly for all presentations of an expansion (A,~a) by finitely many new constants. We end with a conjecture about the situation when all models of a theory are relatively decidable. 1 Introduction The broad goal of computable model theory is to investigate the effective arXiv:1903.00734v1 [math.LO] 2 Mar 2019 aspects of model theory. Here we will carry out exactly this process with the model-theoretic concept of model completeness. This notion is well-known and has been widely studied in model theory, but to our knowledge there has never been any thorough examination of its implications for computability in structures with the domain ω. We now rectify this omission, and find natural and satisfactory equivalents for the basic notion of model completeness of a first-order theory. Our two principal results are each readily stated: that a ∗The second author was supported by NSF grant # DMS-1362206, Simons Foundation grant # 581896, and several PSC-CUNY research awards.