Projective model completeness

1974 ◽  
Vol 39 (1) ◽  
pp. 117-123 ◽  
Author(s):  
George S. Sacerdote
Keyword(s):  
Model Theory ◽  
Predicate Calculus ◽  
Model Completeness ◽  
New Model ◽  
Projective Model ◽  
New Criteria ◽  
Completeness Theorems

The concept of model completeness has been very useful in model theory. In this article we obtain a new model theoretic tool by “reversing the arrows.” Specifically, model completeness deals with the relations between a model of a theory and its extensions; in this paper we shall be concerned with the relation between a model of a theory and its homomorphic pre-images.This work is based on the intuitive principle that metamathematical theorems about universal sentences of the lower predicate calculus and substructures can be translated in a truth-preserving way to theorems about positive sentences and homomorphic images. From model completeness and the completeness theorems which depend on it, this translation gives new criteria for completeness.Special thanks are due to Professor A. Robinson whose encouragement and suggestions have contributed much to these results.

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.

ON THE MODEL THEORY OF THE LOGARITHMIC FUNCTION IN COMPACT LIE GROUPS

2013 ◽  
Vol 12 (08) ◽  
pp. 1350055
Author(s):  
SONIA L'INNOCENTE ◽  
FRANÇOISE POINT ◽  
CARLO TOFFALORI
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Model Theory ◽  
Effective Model ◽  
Logarithmic Function ◽  
Compact Lie Groups ◽  
Model Completeness ◽  
Schanuel's Conjecture ◽  
Logarithm Function ◽  
Natural Expansion

Given a compact linear Lie group G, we form a natural expansion of the theory of the reals where G and the graph of a logarithm function on G live. We prove its effective model-completeness and decidability modulo a suitable variant of Schanuel's Conjecture.

First steps in intuitionistic model theory

1978 ◽  
Vol 43 (1) ◽  
pp. 3-12 ◽  
Author(s):  
H. de Swart
Keyword(s):  
Model Theory ◽  
Infinite Sequence ◽  
Finite Sequence ◽  
Kripke Models ◽  
Disjunction Property ◽  
Completeness Proof ◽  
New Model ◽  
Continuity Principle ◽  
Countably Infinite

In this paper we will do some model theory with respect to the models, defined in [7] and, as in [7], we will work again in intuitionistic metamathematics.In this paper we will only consider models M = ‹S, TM›, where S is one fixed spreadlaw for all models M, namely the universal spreadlaw. That we can restrict ourselves to this class of models is a consequence of the completeness proof, which is sketched in [7, §3].The main tools in this paper will be two model-constructions:(i) In §1 we will consider, under a certain condition C(M0, M, s), the construction of a model R(M0, M, s) from two models M0 and M with respect to the finite sequence s.(ii) In §2 we will construct from an infinite sequence M0, M1, M2, … of models a new model Σi∈INMi.Syntactic proofs of the disjunction property and the explicit definability theorem are well known.C. Smorynski [8] gave semantic proofs of these theorems with respect to Kripke models, however using classical metamathematics. In §1 we will give intuitionistically correct, semantic proofs with respect to the models, defined in [7] using Brouwer's continuity principle.Let W be the fan of all models (see [7, Theorem 2.7]) and let Γ be a countably infinite sequence of sentences.

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.

A THREE-VALUED QUANTIFIED ARGUMENT CALCULUS: DOMAIN-FREE MODEL-THEORY, COMPLETENESS, AND EMBEDDING OF FOL

2017 ◽  
Vol 10 (3) ◽  
pp. 549-582 ◽  
Author(s):  
RAN LANZET
Keyword(s):  
Natural Language ◽  
Model Theory ◽  
Predicate Calculus ◽  
Extended Version ◽  
First Order ◽  
Completeness Result ◽  
Free Model ◽  
Simplifying Assumptions ◽  
Syntactic Restrictions ◽  
Order Predicate Calculus

AbstractThis paper presents an extended version of the Quantified Argument Calculus (Quarc). Quarc is a logic comparable to the first-order predicate calculus. It employs several nonstandard syntactic and semantic devices, which bring it closer to natural language in several respects. Most notably, quantifiers in this logic are attached to one-place predicates; the resulting quantified constructions are then allowed to occupy the argument places of predicates. The version presented here is capable of straightforwardly translating natural-language sentences involving defining clauses. A three-valued, model-theoretic semantics for Quarc is presented. Interpretations in this semantics are not equipped with domains of quantification: they are just interpretation functions. This reflects the analysis of natural-language quantification on which Quarc is based. A proof system is presented, and a completeness result is obtained. The logic presented here is capable of straightforward translation of the classical first-order predicate calculus, the translation preserving truth values as well as entailment. The first-order predicate calculus and its devices of quantification can be seen as resulting from Quarc on certain semantic and syntactic restrictions, akin to simplifying assumptions. An analogous, straightforward translation of Quarc into the first-order predicate calculus is impossible.

Model-Completeness and Elementary Properties of Torsion Free Abelian Groups

1974 ◽  
Vol 26 (4) ◽  
pp. 829-840
Author(s):  
Elias Zakon
Keyword(s):  
High Power ◽  
Abelian Groups ◽  
Elementary Theory ◽  
Complete Classification ◽  
Formal Logic ◽  
Predicate Calculus ◽  
Symbolic Logic ◽  
Model Completeness ◽  
Free Abelian Groups ◽  
Torsion Free

The decidability of the elementary theory of abelian groups, and their complete classification by elementary properties (i.e. those formalizable in the lower predicate calculus (LPC) of formal logic), were established by W. Szmielew [13]. More general results were proved by Eklof and Fischer [2], and G. Sabbagh [12]. The rather formidable "high-power" techniques used in obtaining these remarkable results, and the length of the proofs (W. Szmielew's proof takes about 70 pages) triggered off several attempts at simplification. M. I. Kargapolov's proof [3] unfortunately turned out to be erroneous (cf. J. Mennicke's review in the Journal of Symbolic Logic, vol. 32, p. 535).

On the model theory of denumerably long formulas with finite strings of quantifiers

1969 ◽  
Vol 34 (3) ◽  
pp. 437-459 ◽  
Author(s):  
M. Makkai
Keyword(s):  
Model Theory ◽  
Predicate Calculus

In this paper we prove infinitary analogues of model-theoretic results known for finitary logic. The infinitary language we deal with is Lω1ω which is roughly described by saying that, in addition to the usual formation rules of the lower predicate calculus with identity, also the formation of the conjunction and disjunction of countably many formulas is allowed.

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