Affiliations: Research Institute for Fundamental Sciences (RIFS), University of Tabriz, P.O.Box 51666–16471, Bahman 29th Boulevard, Tabriz, IRAN; School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O.Box 19395–5746, Niavaran, Tehran, IRAN. [email protected], [email protected]
Correspondence:
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This research was partially supported by a grant from 𝕀ℙ𝕄 (No. 91030033). Address for correspondence: Research Institute for Fundamental Sciences RIFS, University of Tabriz, P.O.Box 51666–16471, Bahman 29th Boulevard, Tabriz, IRAN.
Abstract: The multiplicative theory of a set of numbers (which could be natural, integer, rational, real or complex numbers) is the first-order theory of the structure of that set with (solely) the multiplication operation (that set is taken to be multiplicative, i.e., closed under multiplication). In this paper we study the multiplicative theories of the complex, real and (positive) rational numbers. These theories (and also the multiplicative theories of natural and integer numbers) are known to be decidable (i.e., there exists an algorithm that decides whether a given sentence is derivable form the theory); here we present explicit axiomatizations for them and show that they are not finitely axiomatizable. For each of these sets (of complex, real and [positive] rational numbers) a language, including the multiplication operation, is introduced in a way that it allows quantifier elimination (for the theory of that set).
