# On decidability and axiomatizability of some ordered structures

On decidability and axiomatizability of some ordered structures The ordered structures of natural, integer, rational and real numbers are studied here. It is known that the theories of these numbers in the language of order are decidable and ﬁnitely axiomatizable. Also, their theories in the language of order and addition are decidable and inﬁnitely axiomatizable. For the language of order and multiplication, it is known that the theories of N and Z are not decidable (and so not axiomatizable by any computably enumerable set of sentences). By Tarski’s theorem, the multiplicative ordered structure of R is decidable also; here we prove this result directly and present an axiomatization. The structure of Q in the language of order and multiplication seems to be missing in the literature; here we show the decidability of its theory by the technique of quantiﬁer elimination, and after presenting an inﬁnite axiomatization for this structure, we prove that it is not ﬁnitely axiomatizable. Keywords Decidability · Undecidability · Completeness · Incompleteness · First-order theory · Quantiﬁer elimination · Ordered structures 1 Introduction and preliminaries (2017); for another proof see Boolos et al. (2007, Theorem 11.2). However, by Gödel’s completeness theorem, the set of Entscheidungsproblem, one of the fundamental problems of logically valid formulas http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Soft Computing Springer Journals

# On decidability and axiomatizability of some ordered structures

, Volume OnlineFirst – Jun 5, 2018
12 pages

/lp/springer_journal/on-decidability-and-axiomatizability-of-some-ordered-structures-ga0Tx3z6Cz
Publisher
Springer Journals
Subject
Engineering; Computational Intelligence; Artificial Intelligence (incl. Robotics); Mathematical Logic and Foundations; Control, Robotics, Mechatronics
ISSN
1432-7643
eISSN
1433-7479
D.O.I.
10.1007/s00500-018-3247-1
Publisher site
See Article on Publisher Site

### Abstract

The ordered structures of natural, integer, rational and real numbers are studied here. It is known that the theories of these numbers in the language of order are decidable and ﬁnitely axiomatizable. Also, their theories in the language of order and addition are decidable and inﬁnitely axiomatizable. For the language of order and multiplication, it is known that the theories of N and Z are not decidable (and so not axiomatizable by any computably enumerable set of sentences). By Tarski’s theorem, the multiplicative ordered structure of R is decidable also; here we prove this result directly and present an axiomatization. The structure of Q in the language of order and multiplication seems to be missing in the literature; here we show the decidability of its theory by the technique of quantiﬁer elimination, and after presenting an inﬁnite axiomatization for this structure, we prove that it is not ﬁnitely axiomatizable. Keywords Decidability · Undecidability · Completeness · Incompleteness · First-order theory · Quantiﬁer elimination · Ordered structures 1 Introduction and preliminaries (2017); for another proof see Boolos et al. (2007, Theorem 11.2). However, by Gödel’s completeness theorem, the set of Entscheidungsproblem, one of the fundamental problems of logically valid formulas

### Journal

Soft ComputingSpringer Journals

Published: Jun 5, 2018

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