On the Significance of Self-Justifying Axiom Systems from the Perspective of Analytic Tableaux

TL;DR

This paper explores self-justifying axiom systems using analytic tableaux, introducing new theorems that improve formal systems' efficiency and discussing their practical significance despite limitations from the Incompleteness Theorem.

Contribution

It presents new theorems for self-justifying systems, including transforming infinite systems into finite ones, advancing the understanding of self-justification in formal logic.

Findings

Finite axiomatizations of self-verifying systems

New theorems enabling tighter formal systems

Insights into the practical utility of self-justification

Abstract

This article will be a continuation of our research into self-justifying systems. It will introduce several new theorems and their applications. (One of these results will transform our previous infinite-sized self-verifying formalisms into tighter systems, with only a finite number of axioms.) It will explain how self-justification is useful, even when the Incompleteness Theorem limits its reach.