当前位置: X-MOL 学术Math. Logic Q. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Expansions of the p ‐adic numbers that interpret the ring of integers
Mathematical Logic Quarterly ( IF 0.3 ) Pub Date : 2020-03-01 , DOI: 10.1002/malq.201900040
Nathanaël Mariaule 1
Affiliation  

Let $\widetilde{\mathbb{Q}_p}$ be the field of $p$-adic numbers in the language of rings. In this paper we consider the theory of $\widetilde{\mathbb{Q}_p}$ expanded by two predicates interpreted by multiplicative subgroups $\alpha^\mathbb{Z}$ and $\beta^\mathbb{Z}$ where $\alpha, \beta\in\mathbb{N}$ are multiplicatively independent. We show that the theory of this structure interprets Peano arithmetic if $\alpha$ and $\beta$ have positive $p$-adic valuation. If either $\alpha$ or $\beta$ has zero valuation we show that the theory of $(\widetilde{\mathbb{Q}_p}, \alpha^\mathbb{Z}, \beta^\mathbb{Z})$ does not interpret Peano arithmetic. In that case we also prove that the theory is decidable iff the theory of $(\widetilde{\mathbb{Q}_p}, \alpha^\mathbb{Z}\cdot \beta^\mathbb{Z})$ is decidable.

中文翻译:

解释整数环的 p-adic 数的展开

令 $\widetilde{\mathbb{Q}_p}$ 是环语言中 $p$-adic 数的域。在本文中,我们考虑了由乘法子群 $\alpha^\mathbb{Z}$ 和 $\beta^\mathbb{Z}$ 解释的两个谓词扩展的 $\widetilde{\mathbb{Q}_p}$ 理论,其中$\alpha, \beta\in\mathbb{N}$ 是乘法独立的。我们表明,如果 $\alpha$ 和 $\beta$ 具有正的 $p$-adic 估值,则该结构的理论可以解释 Peano 算术。如果 $\alpha$ 或 $\beta$ 的估值为零,我们证明 $(\widetilde{\mathbb{Q}_p}, \alpha^\mathbb{Z}, \beta^\mathbb{Z} )$ 不解释皮亚诺算术。在这种情况下,我们还证明了该理论是可判定的,如果 $(\widetilde{\mathbb{Q}_p}, \alpha^\mathbb{Z}\cdot \beta^\mathbb{Z})$ 是可判定的.
更新日期:2020-03-01
down
wechat
bug