论文标题
乘法和数字秩序理论的可决定性
Decidability of the Multiplicative and Order Theory of Numbers
论文作者
论文摘要
本文研究了自然，整数，有理数和实数的有序结构。这些数字的理论在秩序的语言中是可决定的，并且可以有限的公理化。同样，他们以秩序和加法语言的理论是可决定的，并且是无限的公理。对于订单和乘法的语言，众所周知，$ \ mathbb {n} $和$ \ mathbb {z} $的理论是无法决定的（因此，任何可计算的句子集可以通过任何可计算的句子集）。 tarski的定理，$ \ mathbb {r} $的乘法有序结构也可以决定。在本文中，我们直接通过消除量词来证明这一结果，并提出明确的无限公理化。文献中似乎缺少$ \ mathbb {q} $的结构。我们通过消除量词的技术表明了其理论的可决定性，并在为该结构提供了无限的公理化后，我们证明它不是有限的公理。
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.
