Tarski's Undefinability Theorem and Diagonal Lemma

TL;DR

This paper establishes the equivalence between semantic and syntactic versions of Tarski's Undefinability Theorem and the Diagonal Lemma, providing new proofs and linking these foundational results to G"odel's Incompleteness Theorem.

Contribution

It demonstrates the equivalence of semantic and syntactic forms of Tarski's theorem and the Diagonal Lemma, and connects these to G"odel's Incompleteness Theorem with new proof approaches.

Findings

Semantic Tarski's theorem equivalent to Diagonal Lemma

Syntactic Tarski's theorem equivalent to weak diagonal lemma

Syntactic Tarski's theorem implies G"odel-Rosser's Incompleteness

Abstract

We prove the equivalence of the semantic version of Tarski's theorem on the undefinability of truth with a semantic version of the Diagonal Lemma, and also show the equivalence of syntactic Tarski's Undefinability Theorem with a weak syntactic diagonal lemma. We outline two seemingly diagonal-free proofs for these theorems from the literature, and show that syntactic Tarski's theorem can deliver G\"odel-Rosser's Incompleteness Theorem.