The central result of this paper is the small-is-very-small principle for restricted sequential theories. The principle says roughly that whenever the given theory shows that a property has a small witness, i.e. a witness in every definable cut, then it shows that the property has a very small witness: i.e. a witness below a given standard number. We draw various consequences from the central result. For example (in rough formulations): (i) Every restricted, recursively enumerable sequential theory has a finitely axiomatized extension that is conservative w.r.t. formulas of complexity $\leq n$. (ii) Every sequential model has, for any $n$, an extension that is elementary for formulas of complexity $\leq n$, in which the intersection of all definable cuts is the natural numbers. (iii) We have reflection for $\Sigma^0_2$-sentences with sufficiently small witness in any consistent restricted theory $U$. (iv) Suppose $U$ is recursively enumerable and sequential. Suppose further that every recursively enumerable and sequential $V$ that locally inteprets $U$, globally interprets $U$. Then, $U$ is mutually globally interpretable with a finitely axiomatized sequential theory. The paper contains some careful groundwork developing partial satisfaction predicates in sequential theories for the complexity measure depth of quantifier alternations.

中文翻译：

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小就是非常小的原则

本文的核心结果是受限序列理论的小就是非常小原则。该原则粗略地说，每当给定的理论表明一个财产有一个小的见证人，即在每个可定义的切割中都有一个见证人，那么它表明该财产有一个非常小的见证人：即低于给定标准数的见证人。我们从中心结果得出各种结果。例如（在粗略的公式中）：（i）每个受限的、递归可枚举的序列理论都有一个有限公理化的扩展，它是复杂性 $\leq n$ 的保守公式。(ii) 对于任何 $n$，每个序列模型都有一个扩展，对于复杂性 $\leq n$ 的公式是基本的，其中所有可定义割的交集是自然数。(iii) 在任何一致的受限理论 $U$ 中，我们对 $\Sigma^0_2$-sentences 有足够小的见证。(iv) 假设 $U$ 是递归可枚举和顺序的。进一步假设每个局部解释 $U$ 的递归可枚举和顺序 $V$，全局解释 $U$。然后，$U$ 可以通过有限公理化的序列理论相互全局解释。该论文包含一些精心的基础工作，在序列理论中为量词交替的复杂性度量深度开发了部分满意谓词。$U$ 可以通过有限公理化的序列理论相互全局解释。该论文包含一些精心的基础工作，在序列理论中为量词交替的复杂性度量深度开发了部分满意谓词。$U$ 可以通过有限公理化的序列理论相互全局解释。该论文包含一些精心的基础工作，在序列理论中为量词交替的复杂性度量深度开发了部分满意谓词。