Generalizations of the Compactness Theorem and Gödel’s Completeness Theorem for Nonstandard Finite Structures

2007 ◽  
pp. 13-33 ◽  
Author(s):  
Miklós Ajtai
Keyword(s):  
Completeness Theorem ◽  
Compactness Theorem ◽  
Finite Structures ◽  
Gödel’S Completeness Theorem

Compactness, infinitesimals, and the reals

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Real Number ◽  
Model Theory ◽  
Positive Real Number ◽  
Completeness Theorem ◽  
Compactness Theorem ◽  
Positive Real ◽  
Standard Analysis ◽  
Absolute Value ◽  
Gödel’S Completeness Theorem ◽  
Non Standard Analysis

One of the most famous philosophical applications of model theory is Robinson’s attempt to salvage infinitesimals. Infinitesimals are quantities whose absolute value is smaller than that of any given positive real number. Robinson used his non-standard analysis to formalize and vindicate the Leibnizian approach to the calculus. Against this, the historian Bos has questioned whether the infinitesimals of Robinson's non-standard analysis have the same structure as those of Leibniz. We offer a response to Bos, by building valuations into Robinson's non-standard analysis. This chapter also introduces some related discussions of independent interest (compactness, instrumentalism, and o-minimality) and contains a proof of The Compactness Theorem and Gödel’s Completeness Theorem.

scholarly journals Interpretability over peano arithmetic

1999 ◽  
Vol 64 (4) ◽  
pp. 1407-1425
Author(s):  
Claes Strannegård
Keyword(s):  
Modal Logic ◽  
Completeness Theorem ◽  
Peano Arithmetic ◽  
Embedding Theorem ◽  
Compactness Theorem ◽  
Partial Answer ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Interpretability Logic

AbstractWe investigate the modal logic of interpretability over Peano arithmetic. Our main result is a compactness theorem that extends the arithmetical completeness theorem for the interpretability logic ILMω. This extension concerns recursively enumerable sets of formulas of interpretability logic (rather than single formulas). As corollaries we obtain a uniform arithmetical completeness theorem for the interpretability logic ILM and a partial answer to a question of Orey from 1961. After some simplifications, we also obtain Shavrukov's embedding theorem for Magari algebras (a.k.a. diagonalizable algebras).

Uniform inductive definability and infinitary languages

1976 ◽  
Vol 41 (1) ◽  
pp. 109-120
Author(s):  
Anders M. Nyberg
Keyword(s):  
Proof Theory ◽  
Completeness Theorem ◽  
Recursion Theory ◽  
Compactness Theorem ◽  
Consequence Relation ◽  
Admissible Sets ◽  
Inductive Definitions ◽  
Natural Connection ◽  
Recent Approach ◽  
Compactness Theorems

Introduction. The purpose of this paper is to show how results from the theory of inductive definitions can be used to obtain new compactness theorems for uncountable admissible languages. These will include improvements of the compactness theorem by J. Green [9].In [2] Barwise studies admissible sets satisfying the Σ1-compactness theorem. Our approach is to consider admissible sets satisfying what could be called the abstract extended completeness theorem, that is, sets where the consequence relation of the admissible fragment LA is Σ1-definable over A. We will call such sets Σ1-complete. For countable admissible sets, Σ1-completeness follows from the completeness theorem for LA.Having restricted our attention to Σ1-complete sets we are led to a stronger notion also true on countable admissible sets, namely what we shall call uniform Σ1-completeness. We will see that this notion can be viewed as extending to uncountable admissible sets, properties related to both the “recursion theory” and “proof theory” of countable admissible sets.By following Barwise's recent approach to admissible sets allowing “urelements,” we show that there is a natural connection between certain structures arising from the theory of inductive definability, and uniformly Σ1-complete admissible sets . The structures we have in mind are called uniform Kleene structures.

The Discovery of My Completeness Proofs

1996 ◽  
Vol 2 (2) ◽  
pp. 127-158 ◽  
Author(s):  
Leon Henkin
Keyword(s):  
Model Theory ◽  
Scientific Work ◽  
Completeness Theorem ◽  
Large Classes ◽  
Mathematical Structures ◽  
First Order ◽  
Kurt Gödel ◽  
Gödel’S Completeness Theorem ◽  
The Right ◽  
Mathematical Discovery

§1. Introduction. This paper deals with aspects of my doctoral dissertation which contributed to the early development of model theory. What was of use to later workers was less the results of my thesis, than the method by which I proved the completeness of first-order logic—a result established by Kurt Gödel in his doctoral thesis 18 years before.The ideas that fed my discovery of this proof were mostly those I found in the teachings and writings of Alonzo Church. This may seem curious, as his work in logic, and his teaching, gave great emphasis to the constructive character of mathematical logic, while the model theory to which I contributed is filled with theorems about very large classes of mathematical structures, whose proofs often by-pass constructive methods.Another curious thing about my discovery of a new proof of Gödel's completeness theorem, is that it arrived in the midst of my efforts to prove an entirely different result. Such “accidental” discoveries arise in many parts of scientific work. Perhaps there are regularities in the conditions under which such “accidents” occur which would interest some historians, so I shall try to describe in some detail the accident which befell me.A mathematical discovery is an idea, or a complex of ideas, which have been found and set forth under certain circumstances. The process of discovery consists in selecting certain input ideas and somehow combining and transforming them to produce the new output ideas. The process that produces a particular discovery may thus be represented by a diagram such as one sees in many parts of science; a “black box” with lines coming in from the left to represent the input ideas, and lines going out to the right representing the output. To describe that discovery one must explain what occurs inside the box, i.e., how the outputs were obtained from the inputs.

Adding propositional connectives to countable infinitary logic

1975 ◽  
Vol 77 (1) ◽  
pp. 1-6
Author(s):  
Harvey Friedman
Keyword(s):  
Completeness Theorem ◽  
Predicate Calculus ◽  
Compactness Theorem ◽  
Functional Completeness ◽  
Infinitary Logic ◽  
Completeness Theorems ◽  
Infinite Sequences

AbstractFor countable admissible α, one can add a new infinitary propositional connective to so that the extended language obeys the Barwise compactness theorem, and the set of valid sentences is complete α-r.e.Aside from obeying the compactness theorem and a completeness theorem, ordinary finitary predicate calculus is also truth-functionally complete.In (1), Barwise shows that for countable admissible A, provides a fragment of which obeys a compactness theorem and a completeness theorem. However, we of course lose truth-functional completeness, with respect to infinitary propositional connectives that operate on infinite sequences of propositional variables. This raises the question of studying extensions of the language obtained by adding infinitary propositional connectives, in connexion with the Barwise compactness and completeness theorems, and other metatheorems, proved for Some aspects of this project are proposed in (3). It is the purpose of this paper to answer a few of the more basic questions which arise in this connexion.We have not attempted to study the preservation of interpolation or implicit definability. This could be quite interesting if done systematically.

Prerequisites

1993 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Completeness Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
Natural Numbers ◽  
Logical Validity ◽  
First Order ◽  
Gödel’S Completeness Theorem ◽  
Logical Connectives ◽  
Incompleteness Theorems ◽  
Individual Variables

As we remarked in the preface, although this volume is a sequel to our earlier volume G.I.T. (Gödel’s Incompleteness Theorems), it can be read independently by those readers familiar with at least one proof of Gödel’s first incompleteness theorem. In this chapter we give the notation, terminology and main results of G.I.T. that are needed for this volume. Readers familiar with G.I.T. can skip this chapter or perhaps glance through it briefly as a refresher. §0. Preliminaries. we assume the reader to be familiar with the basic notions of first-order logic—the logical connectives, quantifiers, terms, formulas, free and bound occurrences of variables, the notion of interpretations (or models), truth under an interpretation, logical validity (truth under all interpretations), provability (in some complete system of first-order logic with identity) and its equivalence to logical validity (Gödel’s completeness theorem). we let S be a system (theory) couched in the language of first-order logic with identity and with predicate and/or function symbols and with names for the natural numbers. A system S is usually presented by taking some standard axiomatization of first-order logic with identity and adding other axioms called the non-logical axioms of S.we associate with each natural number n an expression n̅ of S called the numeral designating n (or the name of n).we could, for example, take 0̅,1̅,2̅, . . . ,to be the expressions 0,0', 0",..., as we did in G.I.T. we have our individual variables arranged in some fixed infinite sequence v1, v2,..., vn , . . . . By F(v1, ..., vn) we mean any formula whose free variables are all among v1,... ,vn, and for any (natural) numbers k1,...,kn by F(к̅1 ,... к̅n), we mean the result of substituting the numerals к̅1 ,... к̅n, for all free occurrences of v1,... ,vn in F respectively.

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