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05-06-2018 | Foundations | Issue 11/2019

Soft Computing 11/2019

On decidability and axiomatizability of some ordered structures

Journal:
Soft Computing > Issue 11/2019
Authors:
Ziba Assadi, Saeed Salehi
Important notes
Communicated by A. Di Nola.
This is a part of the Ph.D. thesis of the first author written under the supervision of the second author who is partially supported by Grant \(\mathsf{N}^\mathsf{o}\) 90030053 of the Institute for Research in Fundamental Sciences (\(\mathbb {IPM}\)), Tehran, Iran.

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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 \(\mathbb {N}\) and \(\mathbb {Z}\) are not decidable (and so not axiomatizable by any computably enumerable set of sentences). By Tarski’s theorem, the multiplicative ordered structure of \(\mathbb {R}\) is decidable also; here we prove this result directly and present an axiomatization. The structure of \(\mathbb {Q}\) 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.

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