乘法的可判定性与数论的序
Decidability of the Multiplicative and Order Theory of Numbers
摘要：
本文研究了自然数、整数、有理数和实数的序结构。这些数的理论在序的语言中是可判定的和可公理化的。此外，他们的理论在语言的秩序和加法是决定性的和无限公理化。对于顺序和乘法的语言，已知$\mathbb{N}$和$\mathbb{Z}$的理论是不可判定的（因此不能通过任何可计算的可判定的句子集来公理化）。根据Tarski定理，$\mathbb{R}$的乘法序结构也是可判定的.在本文中，我们直接用量词消去法证明了这个结果，并给出了一个显式的无穷公理化。文献中似乎缺少顺序和乘法语言中的$\mathbb{Q}$的结构。我们证明了它的理论的可判定性的量词消除的技术，并提出了一个无限公理化后，这个结构，我们证明了它是不可公理化的。
英文摘要：
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.
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