Implicit Commitment in a General Setting

TL;DR

This paper explores the concept of implicit commitment in formal systems, generalizing previous work by examining iterative implicit commitments and extending analysis to theories in arbitrary first-order languages.

Contribution

It broadens the framework of implicit commitments by introducing iterations and applying the concept to theories beyond arithmetic.

Findings

Generalized implicit commitment framework to multiple iterations.

Extended analysis to theories in arbitrary first-order languages.

Identified open questions for future research.

Abstract

G\"odel'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, \L elyk 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.