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.

First-order theories

2004 ◽  
Author(s):  
Shawn Hedman
Keyword(s):  
Model Theory ◽  
Quantifier Elimination ◽  
Completeness Theorem ◽  
Order Theory ◽  
Linear Orders ◽  
Mathematical Structures ◽  
First Order ◽  
Table Model ◽  
Points Of View ◽  
The Relationship

We continue our study of Model Theory. This is the branch of logic concerned with the interplay between sentences of a formal language and mathematical structures. Primarily, Model Theory studies the relationship between a set of first-order sentences T and the class Mod(T) of structures that model T. Basic results of Model Theory were proved in the previous chapter. For example, it was shown that, in first-order logic, every model has a theory and every theory has a model. Put another way, T is consistent if and only if Mod(T) is nonempty. As a consequence of this, we proved the Completeness theorem. This theorem states that T ├ φ if and onlyif M ╞ φ for each M in Mod(T). So to study a theory T, we can avoid the concept of ├ and the methods of deduction introduced in Chapter 3, and instead work with the concept of ╞ and analyze the class Mod(T). More generally, we can go back and forth between the notions on the left side of the following table and their counterparts on the right. Progress in mathematics is often the result of having two or more points of view that are shown to be equivalent. A prime example is the relationship between the algebra of equations and the geometry of the graphs defined by the equations. Combining these two points of view yield concepts and results that would not be possible in either geometry or algebra alone. The Completeness theorem equates the two points of view exemplified in the above table. Model Theory exploits the relationship between these two points of view to investigate mathematical structures. First-order theories serve as our objects of study in this chapter. A first-order theory may be viewed as a consistent set of sentences T or as an elementary class of structures Mod(T). We shall present examples of theories and consider properties that the theories mayor may not possess such as completeness, categoricity, quantifier-elimination, and model-completeness. The properties that a theory possesses shed light on the structures that model the theory. We analyze examples of first-order structures including linear orders, vector spaces, the random graph, and the complex numbers.

The work of Kurt Gödel

1976 ◽  
Vol 41 (4) ◽  
pp. 761-778 ◽  
Author(s):  
Stephen C. Kleene
Keyword(s):  
Functional Calculus ◽  
Completeness Theorem ◽  
Predicate Calculus ◽  
Closed Formula ◽  
Modern Logic ◽  
Von Neumann ◽  
First Order ◽  
Kurt Gödel ◽  
Gödel’S Completeness Theorem ◽  
Order Predicate Calculus

I first heard the name of Kurt Gödel when, as a graduate student at Princeton in the fall of 1931, I attended a colloquium at which John von Neumann was the speaker, von Neumann could have spoken on work of his own; but instead he gave an exposition of Gödel's results of formally undecidable propositions [1931].Today I shall begin with Gödel's paper [1930] on The completeness of the axioms of the functional calculus of logic, or of what we now often call “the first-order predicate calculus”, using “predicate” as synonymous with “propositional function”.Alonzo Church wrote ([1944, p. 62] and [1956, pp. 288–289]), “the first explicit formulation of the functional calculus of first order as an independent logistic system is perhaps in the first edition of Hilbert and Ackermann's Grundzüge der theoretischen Logik (1928).” Clearly, this formalism is not complete in the sense that each closed formula or its negation is provable. (A closed formula, or sentence, is a formula without free occurrences of variables.) But Hilbert and Ackermann observe, “Whether the system of axioms is complete at least in the sense that all the logical formulas which are correct for each domain of individuals can actually be derived from them is still an unsolved question.” [1928, p. 68].This question Gödel answered in the affirmative in his Ph.D. thesis (Vienna, 1930), of which the paper under discussion is a rewritten version.I shall not describe Gödel's proof. Perhaps no theorem in modern logic has been proved more often than Gödel's completeness theorem for the first-order predicate calculus. It stands at the focus of a complex of fundamental theorems, which different scholars have approached from various directions (e.g. Kleene [1967, Chapter VI]).

On the strong semantical completeness of the intuitionistic predicate calculus

1968 ◽  
Vol 33 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Richmond H. Thomason
Keyword(s):  
Model Theory ◽  
Completeness Theorem ◽  
Predicate Calculus ◽  
First Order ◽  
Semantic Tableaux ◽  
Weak Completeness ◽  
Skolem Theorem ◽  
Completeness Theorems

In Kripke [8] the first-order intuitionjstic predicate calculus (without identity) is proved semantically complete with respect to a certain model theory, in the sense that every formula of this calculus is shown to be provable if and only if it is valid. Metatheorems of this sort are frequently called weak completeness theorems—the object of the present paper is to extend Kripke's result to obtain a strong completeness theorem for the intuitionistic predicate calculus of first order; i.e., we will show that a formula A of this calculus can be deduced from a set Γ of formulas if and only if Γ implies A. In notes 3 and 5, below, we will indicate how to account for identity, as well. Our proof of the completeness theorem employs techniques adapted from Henkin [6], and makes no use of semantic tableaux; this proof will also yield a Löwenheim-Skolem theorem for the modeling.

scholarly journals Notes on Quasiminimality and Excellence

2004 ◽  
Vol 10 (3) ◽  
pp. 334-366 ◽  
Author(s):  
John T. Baldwin
Keyword(s):  
Algebraic Geometry ◽  
Model Theory ◽  
Special Kind ◽  
Order Logic ◽  
First Order Logic ◽  
Structure Theory ◽  
Mathematical Structures ◽  
First Order ◽  
Elementary Class ◽  
Abstract Elementary Class

AbstractThis paper ties together much of the model theory of the last 50 years. Shelah's attempts to generalize the Morley theorem beyond first order logic led to the notion of excellence, which is a key to the structure theory of uncountable models. The notion of Abstract Elementary Class arose naturally in attempting to prove the categoricity theorem for Lω1,ω(Q). More recently, Zilber has attempted to identify canonical mathematical structures as those whose theory (in an appropriate logic) is categorical in all powers. Zilber's trichotomy conjecture for first order categorical structures was refuted by Hrushovski, by the introducion of a special kind of Abstract Elementary Class. Zilber uses a powerful and essentailly infinitary variant on these techniques to investigate complex exponentiation. This not only demonstrates the relevance of Shelah's model theoretic investigations to mainstream mathematics but produces new results and conjectures in algebraic geometry.

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.

Ranked partial structures

2003 ◽  
Vol 68 (4) ◽  
pp. 1109-1144
Author(s):  
Timothy J. Carlson
Keyword(s):  
Model Theory ◽  
Completeness Theorem ◽  
Order Logic ◽  
Basic Theory ◽  
First Order Logic ◽  
Theoretic Method ◽  
First Order ◽  
Partial Structures ◽  
Skolem Theorem

AbstractThe theory of ranked partial structures allows a reinterpretation of several of the standard results of model theory and first-order logic and is intended to provide a proof-theoretic method which allows for the intuitions of model theory. A version of the downward Löwenheim-Skolem theorem is central to our development. In this paper we will present the basic theory of ranked partial structures and their logic including an appropriate version of the completeness theorem.

scholarly journals A proof of completeness for continuous first-order logic

2010 ◽  
Vol 75 (1) ◽  
pp. 168-190 ◽  
Author(s):  
Itaï Ben Yaacov ◽  
Arthur Paul Pedersen
Keyword(s):  
Model Theory ◽  
Completeness Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
Complete Theory ◽  
Strong Completeness ◽  
First Order ◽  
Completeness Result ◽  
Metric Structures ◽  
Natural Classes

AbstractContinuous first-order logic has found interest among model theorists who wish to extend the classical analysis of “algebraic” structures (such as fields, group, and graphs) to various natural classes of complete metric structures (such as probability algebras, Hilbert spaces, and Banach spaces). With research in continuous first-order logic preoccupied with studying the model theory of this framework, we find a natural question calls for attention. Is there an interesting set of axioms yielding a completeness result?The primary purpose of this article is to show that a certain, interesting set of axioms does indeed yield a completeness result for continuous first-order logic. In particular, we show that in continuous first-order logic a set of formulae is (completely) satisfiable if (and only if) it is consistent. From this result it follows that continuous first-order logic also satisfies anapproximatedform of strong completeness, whereby Σ⊧φ(if and) only if Σ⊢φ∸2−nfor alln < ω. This approximated form of strong completeness asserts that if Σ⊧φ, then proofs from Σ, being finite, can provide arbitrarily better approximations of the truth ofφ.Additionally, we consider a different kind of question traditionally arising in model theory—that of decidability. When is the set of all consequences of a theory (in a countable, recursive language) recursive? Say that a complete theoryTisdecidableif for every sentenceφ, the valueφTis a recursive real, and moreover, uniformly computable fromφ. IfTis incomplete, we say it is decidable if for every sentenceφthe real numberφTois uniformly recursive fromφ, whereφTois the maximal value ofφconsistent withT. As in classical first-order logic, it follows from the completeness theorem of continuous first-order logic that if a complete theory admits a recursive (or even recursively enumerable) axiomatization then it is decidable.

Systematization of finite many-valued logics through the method of tableaux

1987 ◽  
Vol 52 (2) ◽  
pp. 473-493 ◽  
Author(s):  
Walter A. Carnielli
Keyword(s):  
Model Theory ◽  
Proof Theory ◽  
Completeness Theorem ◽  
First Order ◽  
Formal Systems ◽  
Skolem Theorem ◽  
Consistency Properties ◽  
Model Existence ◽  
Model Existence Theorem ◽  
Theory And Model

AbstractThis paper presents a unified treatment of the propositional and first-order many-valued logics through the method of tableaux. It is shown that several important results on the proof theory and model theory of those logics can be obtained in a general way.We obtain, in this direction, abstract versions of the completeness theorem, model existence theorem (using a generalization of the classical analytic consistency properties), compactness theorem and Löwenheim-Skolem theorem.The paper is completely self-contained and includes examples of application to particular many-valued formal systems.

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.

PENURUNAN SUHU TUBUH MENGGUNAKAN TEPID WATER SPONGING DENGAN PENDEKATAN KONSERVASI LEVINE

2019 ◽  
Vol 11 (1) ◽  
pp. 33-40
Author(s):  
Muhammad Khabib Burhanuddin Iqomh ◽  
Nani Nurhaeni ◽  
Dessie Wanda
Keyword(s):  
Body Temperature ◽  
Nursing Care ◽  
Human Body ◽  
Model Theory ◽  
Pharmacological Treatment ◽  
Scientific Work ◽  
Pharmacological Therapy ◽  
Nursing Process ◽  
Tepid Water ◽  
Conservation Model

Peningkatan suhu tubuh  menyebabkan rasa tidak nyaman, gelisah pada anak, sehingga waktu untuk istirahat menjadi terganggu.Tatalaksana pada anak dengan demam dapat dilakukan dengan metode farmakologi dan non farmakologi. Tepid water spongingmerupakan tatalaksana non farmakologi. Konservasi adalah serangkaian sistem agar tubuh manusia mampu menjalankan fungsi, beradaptasi untuk melangsungkan kehidupan. Perawat mempunyai peran untuk membantu anak dalam mengatasi gangguan termoregulasi. Karya ilmiah ini bertujuan untuk mengetahui efektifitas penurunan suhu tubuh menggunakan tepid water sponging dengan pendekatanl konservasi Levine di ruang rawat infeksi. Efektifitas diukur dalam pemberian asuhan keperawatan berdasarkan proses keperawatan yang terdapat dalam model konservasi Levine yaitu: pengkajian, menentukan trophicognosis, menentukan hipotesis, intervensi dan evaluasi. Terdapat lima kasus yang dibahas. Hasil penerapan model konservasi Levine mampu meningkatkan kemampuan anak dalam mempertahankan fungsi tubuh dan beradaptasi terhadap perubahan. Kombinasi tepid water sponging dan terapi farmakologi mampu mengatasi demam dengan cepat dibanding terapi farmakologi.   Kata kunci: termoregulasi, tepid water sponging, teori model konservasi Levine   REDUCTION OF BODY TEMPERATURE USING TEPID WATER SPONGINGWITH THE LEVINE CONSERVATION APPROACH   ABSTRACT Increased body temperature causes discomfort, anxiety in children, so that the time to rest becomes disturbed. Management of children with fever can be done by pharmacological and non-pharmacological methods. Tepid water sponging is a non-pharmacological treatment. Conservation is a series of systems so that the human body is able to function, adapt to life. Nurses have a role to help children overcome thermoregulation disorders. This scientific work aims to determine the effectiveness of decreasing body temperature using tepid water sponging with the approach of Levine conservation in the infectious care room. Effectiveness is measured in the provision of nursing care based on the nursing process contained in the Levine conservation model, namely: assessment, determining trophicognosis, determining hypotheses, intervention and evaluation. There are five cases discussed. The results of the application of the Levine conservation model are able to improve the ability of children to maintain body functions and adapt to changes. The combination of tepid water sponging and pharmacological therapy is able to overcome fever quickly compared to pharmacological therapy.   Keywords: thermoregulation, tepid water sponging, Levine conservation model theory  

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