# On decidability and axiomatizability of some ordered structures

## 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.

## Keywords

Decidability Undecidability Completeness Incompleteness First-order theory Quantifier elimination Ordered structures## Notes

### Acknowledgements

The authors gratefully thank the two anonymous referees of the journal for their valuable comments and suggestions.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

### Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

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