Reflecting on Truth.
Reflecting on Truth.
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反思真理。
DOI：
10.1093/logcom/exad025
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发表时间：
2016
影响因子：
0.7
通讯作者：
Graham Emil Leigh
中科院分区：
文献类型：
--
作者：
Graham Emil Leigh
Gödel的不完备性定理表明，没有一个单一的形式系统可以捕捉到一个人的数学信念的全部，同时指出了一个层次结构的系统，其逻辑强度不断增强，使那些隐含的假设越来越明确。这种内隐承诺的概念直接或间接地推动了逻辑和数学基础的几个研究项目；然而，对内隐承诺本身的概念还没有直接的逻辑分析。在最近的一篇论文中，我们通过研究隐性承诺的必要条件对这个项目进行了初步评估；从一个算术系统的隐式承诺的看似薄弱的假设，可以推导出一个一致反射原则-陈述定理的所有数值实例都为真-必须包含在隐式承诺中。这项研究提出了尚未探索的研究途径和开放的问题。本文论述了其中的主要问题。我们在两个维度上推广了隐式承诺的基本框架：根据基本隐式承诺算子的迭代，并通过研究任意一阶语言的理论隐式承诺，而不仅仅是用算术语言表达。
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 implicit assumptions. This notion of 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, we 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, it can be derived that a uniform reflection principle for—stating that all numerical instances of theorems ofare true—must be contained in’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.
真理很简单
影响因子：
1.8
作者：
Horsten L
通讯作者：
Horsten L