Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Decidability of the Multiplicative and Order Theory of Numbers   The ordered structures of natural, integer, rational and real numbers are
studied in this thesis. 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. In this
thesis we prove this result directly by quantifier elimination and present an
explicit infinite axiomatization. The structure of $\mathbb{Q}$ in the language
of order and multiplication seems to be missing in the literature. 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.

11
The categories of this paper are: hep-lat
16
The categories of this paper are: stat
19
The categories of this paper are: eess
15
The categories of this paper are: nucl-ex
31
The categories of this paper are: ao-sci
37
The categories of this paper are: supr-con
24
The categories of this paper are: dg-ga
26
The categories of this paper are: bayes-an
17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
33
The categories of this paper are: plasm-ph
8
The categories of this paper are: hep-ex
34
The categories of this paper are: chem-ph
25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
20
The categories of this paper are: acc-phys