On How the Introducing of a New $\theta$ Function Symbol Into Arithmetic's Formalism Is Germane to Devising Axiom Systems that Can Appreciate Fragments of Their Own Hilbert Consistency

TL;DR

This paper introduces a new $ heta$ function primitive that enhances an axiom system's ability to verify its own consistency and prove certain theorems, advancing self-referential formal systems.

Contribution

It proposes a novel $ heta$ function symbol enabling the construction of an axiom system that can partially verify its own Hilbert consistency and prove key $ ext{Pi}_1$ theorems.

Findings

The $ heta$ function approximates the efficiency of addition, multiplication, and successor operations.

The IQFS(PA+) system can verify a fragment of its own consistency.

The system proves isomorphic counterparts of all $ ext{Pi}_1$ theorems of Peano Arithmetic.

Abstract

A new $\theta$ function primitive is proposed that almost achieves the combined efficiency of the addition, multiplication and successor growth operations. This $\theta$ function symbol enables the constructing of an "IQFS(PA+)" axiom system that can corroborate a fragmentary definition of its own Hilbert consistency, while it will simultaneously verify isomorphic counterparts of all Peano Arithmetic's $\Pi_1$ theorems. Many propositions and intermediate results are also established. Only one intermediate result, which most readers will intuit should be true, does remain formally unproven.