Acceptable Complexity Measures of Theorems

TL;DR

This paper explores alternative measures of theorem complexity that align with the heuristic principle, aiming to better understand the nature and frequency of unprovable true statements in formal systems.

Contribution

It introduces the concept of acceptable complexity measures of theorems and investigates their existence beyond the original measure used in prior work.

Findings

Established the definition of acceptable complexity measures

Proved the existence of multiple measures satisfying the heuristic principle

Extended understanding of theorem complexity beyond traditional measures

Abstract

In 1931, G\"odel presented in K\"onigsberg his famous Incompleteness Theorem, stating that some true mathematical statements are unprovable. Yet, this result gives us no idea about those independent (that is, true and unprovable) statements, about their frequency, the reason they are unprovable, and so on. Calude and J\"urgensen proved in 2005 Chaitin's "heuristic principle" for an appropriate measure: the theorems of a finitely-specified theory cannot be significantly more complex than the theory itself. In this work, we investigate the existence of other measures, different from the original one, which satisfy this "heuristic principle". At this end, we introduce the definition of acceptable complexity measure of theorems.