$\omega$-consistency for Different Arrays of Quantifiers

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.