On the construction of fully interpreted formal languages which posses their truth predicates

TL;DR

This paper presents a recursive construction method for a fully interpreted formal language with its own truth predicate, enabling the language to internally express and interpret its sentences.

Contribution

It introduces a transfinite recursion approach to build a sublanguage with an embedded truth predicate, advancing formal language theory.

Findings

Constructed a sublanguage with a truth predicate

Used transfinite recursion to define truth assignments

Established a fully interpreted language with internal truth

Abstract

We shall construct by ordinary recursion method subsets to the set $D$ of G\"odel numbers of the sentences of a language $\mathcal L$. That language is formed by sentences of a fully interpreted formal language $L$, called an MA language, and sentences containing a monadic predicate letter $T$. From the class of the constructed subsets of $D$ we extract one set $U$ by transfinite recursion method. Interpret those sentences whose G\"odel numbers are in $U$ as true, and their negations as false. These sentences together form an MA language. It is a sublanguage of $\mathcal L$ having $L$ as its sublanguage, and $T$ is its truth predicate.