一般设定下的隐式承诺
Implicit Commitment in a General Setting
AI总结：
本文推广Łelyk和Nicolai的隐式承诺逻辑框架，通过迭代承诺算子并扩展至任意一阶语言理论，分析形式系统隐式承诺的必要条件。
摘要：
Gödel不完备定理表明，任何单一形式系统都无法囊括一个人的全部数学信念，同时指向一个逻辑强度不断增强的系统层级；这些系统使那些隐式假设逐步变得更加明确。这种“隐式承诺”概念直接或间接地推动了逻辑与数学基础中的若干研究纲领；然而，此前尚没有对隐式承诺这一概念本身进行直接的逻辑分析。在最近的一篇论文中，Łelyk和Nicolai通过研究隐式承诺的必要条件，对该计划进行了初步评估：从关于算术系统S的隐式承诺的看似较弱的假设出发，可以推出S的一致反射原理——即断言S定理的所有数值实例都为真——必定包含在S的隐式承诺中。这项研究引出了尚未探索的研究路径和开放问题。本文处理其中的主要问题。我们沿两个维度推广这一关于隐式承诺的基本框架：一是研究基本隐式承诺算子的迭代，二是研究任意一阶语言中理论的隐式承诺，而不仅限于以算术语言表述的理论。
英文摘要：
Gödel's Incompleteness Theorems suggest that no single formal system can capture the entirety of one's mathematical beliefs, while pointing at a hierarchy of systems of increasing logical strength that make progressively more explicit those \emph{implicit} assumptions. This notion of \emph{implicit commitment} motivates directly or indirectly several research programmes in logic and the foundations of mathematics; yet there hasn't been a direct logical analysis of the notion of implicit commitment itself. In a recent paper, Łelyk and Nicolai carried out an initial assessment of this project by studying necessary conditions for implicit commitments; from seemingly weak assumptions on implicit commitments of an arithmetical system $S$, it can be derived that a uniform reflection principle for $S$ -- stating that all numerical instances of theorems of $S$ are true -- must be contained in $S$'s implicit commitments. This study gave rise to unexplored research avenues and open questions. This paper addresses the main ones. We generalize this basic framework for implicit commitments along two dimensions: in terms of iterations of the basic implicit commitment operator, and via a study of implicit commitments of theories in arbitrary first-order languages, not only couched in an arithmetical language.
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