Paul C. Eklof and Edward R. Fisher. The elementary theory of Abelian groups. Annals of mathematical logic, vol. 4 no. 2 (1972), pp. 115–171.

1974 ◽  
Vol 39 (3) ◽  
pp. 603-604
Author(s):  
C. Smorynski
Keyword(s):  
Mathematical Logic ◽  
Abelian Groups ◽  
Elementary Theory

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).

scholarly journals Alfred Tarski and undecidable theories

1986 ◽  
Vol 51 (4) ◽  
pp. 890-898 ◽  
Author(s):  
George F. McNulty
Keyword(s):  
Mathematical Logic ◽  
Set Theory ◽  
Elementary Theory ◽  
Logical Consequence ◽  
Quantifier Elimination ◽  
Equational Theory ◽  
Boolean Algebras ◽  
Decision Problems ◽  
Alfred Tarski ◽  
Elementary Theories

Alfred Tarski identified decidability within various logical formalisms as one of the principal themes for investigation in mathematical logic. This is evident already in the focus of the seminar he organized in Warsaw in 1926. Over the ensuing fifty-five years, Tarski put forth a steady stream of theorems concerning decidability, many with far-reaching consequences. Just as the work of the 1926 seminar reflected Tarski's profound early interest in decidability, so does his last work, A formalization of set theory without variables, a monograph written in collaboration with S. Givant [8−m]. An account of the Warsaw seminar can be found in Vaught [1986].Tarski's work on decidability falls into four broad areas: elementary theories which are decidable, elementary theories which are undecidable, the undecidability of theories of various restricted kinds, and what might be called decision problems of the second degree. An account of Tarski's work with decidable elementary theories can be found in Doner and van den Dries [1987] and in Monk [1986] (for Boolean algebras). Vaught [1986] discusses Tarski's contributions to the method of quantifier elimination. Our principal concern here is Tarski's work in the remaining three areas.We will say that a set of elementary sentences is a theory provided it is closed with respect to logical consequence and we will say that a theory is decidable or undecidable depending on whether it is a recursive or nonrecursive set. The notion of a theory may be restricted in a number of interesting ways. For example, an equational theory is just the set of all universal sentences, belonging to some elementary theory, whose quantifier-free parts are equations between terms.

scholarly journals On the elementary theory of pairs of real closed fields. II

1982 ◽  
Vol 47 (3) ◽  
pp. 669-679 ◽  
Author(s):  
Walter Baur
Keyword(s):  
Power Series ◽  
Abelian Groups ◽  
Elementary Theory ◽  
Predicate Symbol ◽  
Axiom System ◽  
Closed Field ◽  
Algebraic Numbers ◽  
Real Closed Fields ◽  
Order Language ◽  
Closed Fields

Let ℒ be the first order language of field theory with an additional one place predicate symbol. In [B2] it was shown that the elementary theory T of the class of all pairs of real closed fields, i.e., ℒ-structures ‹K, L›, K a real closed field, L a real closed subfield of K, is undecidable.The aim of this paper is to show that the elementary theory Ts of a nontrivial subclass of containing many naturally occurring pairs of real closed fields is decidable (Theorem 3, §5). This result was announced in [B2]. An explicit axiom system for Ts will be given later. At this point let us just mention that any model of Ts, is elementarily equivalent to a pair of power series fields ‹R0((TA)), R1((TB))› where R0 is the field of real numbers, R1 = R0 or the field of real algebraic numbers, and B ⊆ A are ordered divisible abelian groups. Conversely, all these pairs of power series fields are models of Ts.Theorem 3 together with the undecidability result in [B2] answers some of the questions asked in Macintyre [M]. The proof of Theorem 3 uses the model theoretic techniques for valued fields introduced by Ax and Kochen [A-K] and Ershov [E] (see also [C-K]). The two main ingredients are(i) the completeness of the elementary theory of real closed fields with a distinguished dense proper real closed subfield (due to Robinson [R]),(ii) the decidability of the elementary theory of pairs of ordered divisible abelian groups (proved in §§1-4).I would like to thank Angus Macintyre for fruitful discussions concerning the subject. The valuation theoretic method of classifying theories of pairs of real closed fields is taken from [M].

Angus Macintyre. On ω1-categorical theories of abelian groups. Fundamenta mathematicae, vol. 70 (1971), pp. 253–270. - Angus Macintyre. On ω1-categorical theories of fields. Fundamenta mathematicae, vol. 71 (1971), pp. 1–25. - Joachim Reineke. Minimale Gruppen. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 21 (1975), pp. 357–359. - J. T. Baldwin and Jan Saxl. Logical stability in group theory. The journal of the Australian Mathematical Society, vol. 21 ser. A (1976), pp. 267–276. - B. I. Zil'bér. Gruppy i kol'ca, téoriá kotoryh katégorična (Groups and rings with a categorical theory). Fundamenta mathematicae, vol. 95 (1977), pp. 173–188. - Walter Baur, Gregory Cherlin, and Angus Macintyre. Totally categorical groups and rings. Journal of algebra, vol. 57 (1979), pp. 407–440. - Gregory Cherlin. Groups of small Morley rank. Annals of mathematical logic, vol. 17 (1979), pp. 1–28. - G. Cherlin and S. Shelah. Superstable fields and groups. Annals of mathematical logic, vol. 18 (1980), pp. 227–270. - Bruno Poizat. Sous-groupes définissables d 'un groupe stable. The journal of symbolic logic, vol. 46 (1981), pp. 137–146.

1984 ◽  
Vol 49 (1) ◽  
pp. 317-321 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Mathematical Logic ◽  
Group Theory ◽  
Abelian Groups ◽  
Mathematical Society ◽  
Symbolic Logic ◽  
Morley Rank ◽  
Categorical Theory ◽  
Australian Mathematical Society ◽  
Categorical Groups

scholarly journals The elementary theory of abelian groups

1972 ◽  
Vol 4 (2) ◽  
pp. 115-171 ◽  
Author(s):  
Paul C. Eklof ◽  
Edward R. Fischer
Keyword(s):  
Abelian Groups ◽  
Elementary Theory
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