$\omega$-consistency for Different Arrays of Quantifiers

TL;DR

This paper explores the formalization of $ ext{v}$ statements with various quantifier arrays, establishing their equivalence to $ ext{omega}$-consistency statements and developing a truth theory that proves these statements.

Contribution

It introduces a novel approach to formalize $ ext{v}$ statements with different quantifier arrays and shows their equivalence to $ ext{omega}$-consistency within a basis theory.

Findings

Certain quantifier arrays yield consistency statements equivalent to $ ext{omega}$-consistency.

A new theory of truth is constructed that proves all $ ext{omega}$-consistency statements.

The results deepen understanding of the relationship between quantifier structures and consistency.

Abstract

We study the formalized v statement by allowing the occurrence of different arrays of quantifiers in it. We prove that for some specific arrays of quantifiers we get consistency statements that are S-equivalent to the original $\omega$-consistency statement (S denotes the basis theory to develop metamathematics). We end our paper by creating a theory of truth that proves each $\omega$-consistency-statement.