What is it about?
1- The multiplicative theory of the complex numbers, i.e., the first order theory of the structure (C,x), is decidable by Tarski's Theorem. Here we prove this result directly, and present an explicit axiomatization for it. 2- The multiplicative theory of the real numbers, i.e., the first order theory of the structure (R,x), is decidable by Tarski's Theorem. Here we prove this result directly, and present an explicit axiomatization for it. 3- The multiplicative theory of the positive rational numbers, i.e., the first order theory of the structure (Q+,x), is decidable by Mostowski's Theorem. Here we prove this result directly, and present an explicit axiomatization for it.
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Why is it important?
Axiomatizing mathematical theories is an important goal of mathematical logic. Here we have achieved this goal for the multiplicative theories of the complex, real and (positive) rational numbers.
Perspectives
This is the first step of a big project that I have undertaken in axiomatizing mathematical structures. In a sequel paper, with Ziba Assadi, we have axiomatized the multiplicative and order theory of numbers; such as (R;x,<) and (Q;x,<). I am planning to axiomatize more theories, and have a look at the axiomatizations of some structures which are known to be decidable.
Professor Saeed Salehi
Institute for Research in Fundamental Sciences
Read the Original
This page is a summary of: On Axiomatizability of the Multiplicative Theory of Numbers, Fundamenta Informaticae, March 2018, IOS Press,
DOI: 10.3233/fi-2018-1665.
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