A Completeness Theorem for Unrestricted First-Order Languages
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Scott Spaces and the Dcpo Category
SCOTT SPACES AND THE DCPO CATEGORY JORDAN BROWN Abstract. Directed-complete partial orders (dcpo’s) arise often in the study of λ-calculus. Here we investigate certain properties of dcpo’s and the Scott spaces they induce. We introduce a new construction which allows for the canonical extension of a partial order to a dcpo and give a proof that the dcpo introduced by Zhao, Xi, and Chen is well-filtered. Contents 1. Introduction 1 2. General Definitions and the Finite Case 2 3. Connectedness of Scott Spaces 5 4. The Categorical Structure of DCPO 6 5. Suprema and the Waybelow Relation 7 6. Hofmann-Mislove Theorem 9 7. Ordinal-Based DCPOs 11 8. Acknowledgments 13 References 13 1. Introduction Directed-complete partially ordered sets (dcpo’s) often arise in the study of λ-calculus. Namely, they are often used to construct models for λ theories. There are several versions of the λ-calculus, all of which attempt to describe the ‘computable’ functions. The first robust descriptions of λ-calculus appeared around the same time as the definition of Turing machines, and Turing’s paper introducing computing machines includes a proof that his computable functions are precisely the λ-definable ones [5] [8]. Though we do not address the λ-calculus directly here, an exposition of certain λ theories and the construction of Scott space models for them can be found in [1]. In these models, computable functions correspond to continuous functions with respect to the Scott topology. It is thus with an eye to the application of topological tools in the study of computability that we investigate the Scott topology. -
Right Ideals of a Ring and Sublanguages of Science
RIGHT IDEALS OF A RING AND SUBLANGUAGES OF SCIENCE Javier Arias Navarro Ph.D. In General Linguistics and Spanish Language http://www.javierarias.info/ Abstract Among Zellig Harris’s numerous contributions to linguistics his theory of the sublanguages of science probably ranks among the most underrated. However, not only has this theory led to some exhaustive and meaningful applications in the study of the grammar of immunology language and its changes over time, but it also illustrates the nature of mathematical relations between chunks or subsets of a grammar and the language as a whole. This becomes most clear when dealing with the connection between metalanguage and language, as well as when reflecting on operators. This paper tries to justify the claim that the sublanguages of science stand in a particular algebraic relation to the rest of the language they are embedded in, namely, that of right ideals in a ring. Keywords: Zellig Sabbetai Harris, Information Structure of Language, Sublanguages of Science, Ideal Numbers, Ernst Kummer, Ideals, Richard Dedekind, Ring Theory, Right Ideals, Emmy Noether, Order Theory, Marshall Harvey Stone. §1. Preliminary Word In recent work (Arias 2015)1 a line of research has been outlined in which the basic tenets underpinning the algebraic treatment of language are explored. The claim was there made that the concept of ideal in a ring could account for the structure of so- called sublanguages of science in a very precise way. The present text is based on that work, by exploring in some detail the consequences of such statement. §2. Introduction Zellig Harris (1909-1992) contributions to the field of linguistics were manifold and in many respects of utmost significance. -
Cantor-Von Neumann Set-Theory Fa Muller
Logique & Analyse 213 (2011), x–x CANTOR-VON NEUMANN SET-THEORY F.A. MULLER Abstract In this elementary paper we establish a few novel results in set the- ory; their interest is wholly foundational-philosophical in motiva- tion. We show that in Cantor-Von Neumann Set-Theory, which is a reformulation of Von Neumann's original theory of functions and things that does not introduce `classes' (let alone `proper classes'), developed in the 1920ies, both the Pairing Axiom and `half' the Axiom of Limitation are redundant — the last result is novel. Fur- ther we show, in contrast to how things are usually done, that some theorems, notably the Pairing Axiom, can be proved without invok- ing the Replacement Schema (F) and the Power-Set Axiom. Also the Axiom of Choice is redundant in CVN, because it a theorem of CVN. The philosophical interest of Cantor-Von Neumann Set- Theory, which is very succinctly indicated, lies in the fact that it is far better suited than Zermelo-Fraenkel Set-Theory as an axioma- tisation of what Hilbert famously called Cantor's Paradise. From Cantor one needs to jump to Von Neumann, over the heads of Zer- melo and Fraenkel, and then reformulate. 0. Introduction In 1928, Von Neumann published his grand axiomatisation of Cantorian Set- Theory [1925; 1928]. Although Von Neumann's motivation was thoroughly Cantorian, he did not take the concept of a set and the membership-relation as primitive notions, but the concepts of a thing and a function — for rea- sons we do not go into here. This, and Von Neumann's cumbersome nota- tion and terminology (II-things, II.I-things) are the main reasons why ini- tially his theory remained comparatively obscure. -
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. -
Contents 3 Homomorphisms, Ideals, and Quotients
Ring Theory (part 3): Homomorphisms, Ideals, and Quotients (by Evan Dummit, 2018, v. 1.01) Contents 3 Homomorphisms, Ideals, and Quotients 1 3.1 Ring Isomorphisms and Homomorphisms . 1 3.1.1 Ring Isomorphisms . 1 3.1.2 Ring Homomorphisms . 4 3.2 Ideals and Quotient Rings . 7 3.2.1 Ideals . 8 3.2.2 Quotient Rings . 9 3.2.3 Homomorphisms and Quotient Rings . 11 3.3 Properties of Ideals . 13 3.3.1 The Isomorphism Theorems . 13 3.3.2 Generation of Ideals . 14 3.3.3 Maximal and Prime Ideals . 17 3.3.4 The Chinese Remainder Theorem . 20 3.4 Rings of Fractions . 21 3 Homomorphisms, Ideals, and Quotients In this chapter, we will examine some more intricate properties of general rings. We begin with a discussion of isomorphisms, which provide a way of identifying two rings whose structures are identical, and then examine the broader class of ring homomorphisms, which are the structure-preserving functions from one ring to another. Next, we study ideals and quotient rings, which provide the most general version of modular arithmetic in a ring, and which are fundamentally connected with ring homomorphisms. We close with a detailed study of the structure of ideals and quotients in commutative rings with 1. 3.1 Ring Isomorphisms and Homomorphisms • We begin our study with a discussion of structure-preserving maps between rings. 3.1.1 Ring Isomorphisms • We have encountered several examples of rings with very similar structures. • For example, consider the two rings R = Z=6Z and S = (Z=2Z) × (Z=3Z). -
A Remark About Algebraicity in Complete Partial Orders
A remark ab out algebraicity in complete partial orders Draft Leonid Libkin Department of Computer and Information Science University of Pennsylvania Philadelphia PA USA Email libkinsaulcisupennedu Abstract I prove a characterization theorem for algebraic b ounded complete cp os similar to that for algebraic lattices It is wellknown that a lattice is algebraic i it isomorphic to a lattice of subalgebras of an algebra Algebraicity plays the central role in denotational semantics for programming languages but the structures used there are not exactly algebraic lattices they are complete algebraic partial orders In this note I shall characterize such p osets as p osets of certain subalgebras of partial algebras Let me recall the denitions A p oset is called complete and is usually abbreviated as a cpo if it contains least upp er b ounds or suprema of directed subsets I shall use t for supremum An element x of D is called compact if x tX where X D is directed implies x x for some x X A cp o is called algebraic if for any x D the set of compact elements b elow x is directed and its supremum equals x A cp o D is bounded complete if supremum of X D denoted by tX as well exists whenever X is b ounded ab ove in D ie there is a D such that a x for all x X I shall use a more convenient notation a a instead of tfa a g An element x of a b ounded complete cp o D is compact 1 n 1 n if whenever tX exists and x tX x tX where X X is nite In a b ounded complete cp o the set of compact elements b elow any element is always -
An Outline of Algebraic Set Theory
An Outline of Algebraic Set Theory Steve Awodey Dedicated to Saunders Mac Lane, 1909–2005 Abstract This survey article is intended to introduce the reader to the field of Algebraic Set Theory, in which models of set theory of a new and fascinating kind are determined algebraically. The method is quite robust, admitting adjustment in several respects to model different theories including classical, intuitionistic, bounded, and predicative ones. Under this scheme some familiar set theoretic properties are related to algebraic ones, like freeness, while others result from logical constraints, like definability. The overall theory is complete in two important respects: conventional elementary set theory axiomatizes the class of algebraic models, and the axioms provided for the abstract algebraic framework itself are also complete with respect to a range of natural models consisting of “ideals” of sets, suitably defined. Some previous results involving realizability, forcing, and sheaf models are subsumed, and the prospects for further such unification seem bright. 1 Contents 1 Introduction 3 2 The category of classes 10 2.1 Smallmaps ............................ 12 2.2 Powerclasses............................ 14 2.3 UniversesandInfinity . 15 2.4 Classcategories .......................... 16 2.5 Thetoposofsets ......................... 17 3 Algebraic models of set theory 18 3.1 ThesettheoryBIST ....................... 18 3.2 Algebraic soundness of BIST . 20 3.3 Algebraic completeness of BIST . 21 4 Classes as ideals of sets 23 4.1 Smallmapsandideals . .. .. 24 4.2 Powerclasses and universes . 26 4.3 Conservativity........................... 29 5 Ideal models 29 5.1 Freealgebras ........................... 29 5.2 Collection ............................. 30 5.3 Idealcompleteness . .. .. 32 6 Variations 33 References 36 2 1 Introduction Algebraic set theory (AST) is a new approach to the construction of models of set theory, invented by Andr´eJoyal and Ieke Moerdijk and first presented in [16]. -
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 -
A New Proof of the Completeness of the Lukasiewicz Axioms^)
A NEW PROOF OF THE COMPLETENESS OF THE LUKASIEWICZ AXIOMS^) BY C. C. CHANG The purpose of this note is to provide a new proof for the completeness of the Lukasiewicz axioms for infinite valued propositional logic. For the existing proof of completeness and a history of the problem in general we refer the readers to [l; 2; 3; 4]. The proof as was given in [4] was essentially metamathematical in nature; the proof we offer here is essentially algebraic in nature, which, to some extent, justifies the program initiated by the author in [2]. In what follows we assume thorough familiarity with the contents of [2] and adopt the notation and terminology of [2]. The crux of this proof is con- tained in the following two observations: Instead of using locally finite MV- algebras as the basic building blocks in the structure theory of MV-algebras, we shall use linearly ordered ones. The one-to-one correspondence between linearly ordered MV-algebras and segments of ordered abelian groups enables us to make use of some known results in the first-order theory of ordered abelian groups(2). We say that P is a prime ideal of an MV-algebra A if, and only if, (i) P is an ideal of A, and (ii) for each x, yEA, either xyEP or xyEP- Lemma 1. If P is a prime ideal of A, then A/P is a linearly ordered MV- algebra. Proof. By 3.11 of [2], we have to prove that given x/P and y/P, either x/P^y/P or y/P^x/P. -
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. -
More Model Theory Notes Miscellaneous Information, Loosely Organized
More Model Theory Notes Miscellaneous information, loosely organized. 1. Kinds of Models A countable homogeneous model M is one such that, for any partial elementary map f : A ! M with A ⊆ M finite, and any a 2 M, f extends to a partial elementary map A [ fag ! M. As a consequence, any partial elementary map to M is extendible to an automorphism of M. Atomic models (see below) are homogeneous. A prime model of T is one that elementarily embeds into every other model of T of the same cardinality. Any theory with fewer than continuum-many types has a prime model, and if a theory has a prime model, it is unique up to isomorphism. Prime models are homogeneous. On the other end, a model is universal if every other model of its size elementarily embeds into it. Recall a type is a set of formulas with the same tuple of free variables; generally to be called a type we require consistency. The type of an element or tuple from a model is all the formulas it satisfies. One might think of the type of an element as a sort of identity card for automorphisms: automorphisms of a model preserve types. A complete type is the analogue of a complete theory, one where every formula of the appropriate free variables or its negation appears. Types of elements and tuples are always complete. A type is principal if there is one formula in the type that implies all the rest; principal complete types are often called isolated. A trivial example of an isolated type is that generated by the formula x = c where c is any constant in the language, or x = t(¯c) where t is any composition of appropriate-arity functions andc ¯ is a tuple of constants. -
Paradox and Foundation Zach Weber Submitted in Total Fulfilment of The
Paradox and Foundation Zach Weber Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy May 2009 School of Philosophy, Anthropology and Social Inquiry The University of Melbourne This is to certify that - the thesis comprises only my original work towards the PhD, - due acknowledgement has been made in the text to all other material used, - the thesis is less than 100,000 words in length. Preface Dialethic paraconsistency is an approach to formal and philosophical theories in which some but not all contradictions are true. Advancing that program, this thesis is about paradoxes and the foundations of mathematics, and is divided accordingly into two main parts. The first part concerns the history and philosophy of set theory from Cantor through the independence proofs, focusing on the set concept. A set is any col- lection of objects that is itself an object, with identity completely determined by membership. The set concept is called naive because it is inconsistent. I argue that the set concept is inherently and rightly paradoxical, because sets are both intensional and extensional objects: Sets are predicates in extension. All consistent characterizations of sets are either not explanatory or not coherent. To understand sets, we need to reason about them with an appropriate logic; paraconsistent naive set theory is situated as a continuation of the original foundational project. The second part produces a set theory deduced from an unrestricted compre- hension principle using the weak relevant logic DLQ, dialethic logic with quantifiers. I discuss some of the problems involved with embedding in DLQ, especially related to identity and substitution.