Gödel’s Completeness Theorem

2005 ◽  
pp. 127-136
Keyword(s):  
Completeness Theorem ◽  
Gödel’S Completeness Theorem

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.

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.

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.

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