Theorems of Tarski's Undefinability and Godel's Second Incompleteness-Computationally

TL;DR

This paper unifies G"odel's Second Incompleteness Theorem and Tarski's Undefinability of Truth, providing a bi-theoretic version and a relativized perspective linking the two foundational results in logic.

Contribution

It introduces a bi-theoretic derivability condition for G"odel's theorem and unifies it with Tarski's theorem through a relativized framework.

Findings

A bi-theoretic version of G"odel's Second Incompleteness Theorem.

A relativized interpretation of Tarski's Undefinability of Truth.

Unified view connecting the two classical theorems.

Abstract

We present a version of G\"odel's Second Incompleteness Theorem for recursively enumerable consistent extensions of a fixed axiomatizable theory, by incorporating some bi-theoretic version of the derivability conditions. We also argue that Tarski's theorem on the Undefinability of Truth is G\"odel's First Incompleteness Theorem relativized to definable oracles; a unification of these two theorems is given.