含指数运算语言的Shepherdson定理类比
Analogues of Shepherdson's Theorem for a language with exponentiation
AI总结：
本文研究在算术语言中添加指数或幂函数后的开归纳理论，通过实闭域的指数整数部分刻画模型，得到Shepherdson定理的类比，并给出充分条件与独立性结果。
摘要：
1964年，Shepherdson \cite{shepherdson:1964} 证明了一个离散有序半环 $\mathcal{M}^+$ 满足 $\sf{IOpen}$（无量化词归纳）当且仅当相应的环 $\mathcal{M}$ 是 $\mathcal{M}$ 的商域实闭包的一个整数部分。本文考虑在算术语言中添加指数或幂函数后的开归纳模式，并试图找到这些理论模型的类似准则。对于实闭域理论的若干扩张 $T$，我们得到以下意义上的Shepherdson定理类比：如果指数域 $\mathcal R$ 是 $T$ 的模型，且离散有序环 $\mathcal M$ 是 $\mathcal R$ 的一个（指数）整数部分，则 $\mathcal M^+$ 是扩张语言中开归纳的模型。反之，一般而言仍是一个开放问题。然而，我们分离出一个与著名的Bernoulli不等式相关的自然充分条件，在该条件下此结果成立。我们定义了通常开归纳的一个有限扩张 $T$，使得对任意离散有序环 $\mathcal M$，半环 $\mathcal M^+$ 满足 $T$ 当且仅当存在一个指数实闭域 $\mathcal R$ 满足不等式 $\exp(x) \geqslant 1 + x$，且 $\mathcal M$ 是 $\mathcal R$ 的一个指数整数部分。利用这些结果，我们得到了这些理论的一些具体独立性结果。
英文摘要：
In 1964 Shepherdson \cite{shepherdson:1964} proved that a discretely ordered semiring $\mathcal{M}^+$ satisfies $\sf{IOpen}$ (quantifier free induction) iff the corresponding ring $\mathcal{M}$ is an integer part of the real closure of the quotient field of $\mathcal{M}$. In this paper, we consider open induction schema in the language of arithmetic expanded by exponentiation or by the power function and try to find similar criteria for models of these theories. For several expansions $T$ of the theory of real closed fields we obtain analogues of Shepherdson's Theorem in the following sense: If an exponential field $\mathcal R$ is a model of $T$ and a discretely ordered ring $\mathcal M$ is an (exponential) integer part of $\mathcal R$, then $\mathcal M^+$ is a model of the open induction in the expanded language. The proof of the opposite implication, in general, remains an open question. However, we isolate a natural sufficient condition, related to the well-known Bernoulli inequality, under which this result holds. We define a finite extension $T$ of the usual open induction so that, for any discretely ordered ring $\mathcal M$, the semiring $\mathcal M^+$ satisfies $T$ iff there is an exponential real closed field $\mathcal R$ with the inequality $\exp(x) \geqslant 1 + x$ such that $\mathcal M$ is an exponential integer part of $\mathcal R$. Using these results, we obtain some concrete independence results for these theories.
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