关于乘法数论的公理化
On Axiomatizability of the Multiplicative Theory of Numbers
摘要：
一组数（可以是自然数、整数、有理数、实数或复数）的乘法理论是（仅）具有乘法运算的该集合的结构的一阶理论（该集合被认为是乘法的，即，在乘法下闭合）。本文研究复、实、（正）有理数的乘法理论。已知这些理论（以及自然数和整数的乘法理论）是可判定的（即，存在一个算法，决定是否一个给定的句子是可推导的理论）;在这里，我们提出了明确的公理化，他们并表明，他们是不可公理化。对于这些集合（复数，实数和[正]有理数）中的每一个，都引入了一种语言，包括乘法运算，它允许量词消除（对于该集合的理论）。
英文摘要：
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).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
