A reunion of Gödel, Tarski, Carnap and Rosser.

This article focuses on unifying four foundational results in mathematical logic from the early 20th century: Gödel’s first incompleteness theorem (1931), Tarski’s undefinability theorem (1933), Carnap’s diagonal lemma (1934), and Rosser’s strengthening of Gödel’s incompleteness theorem (1936). It establishes that these theorems are equivalent within a sufficiently general syntactic framework, meaning that either all hold or none do, and provides a formal setting where none of them apply. The paper also presents a unified proof approach, including a Chaitin-style argument for Tarski’s theorem and the weak diagonal lemma, and offers a constructive method for the strong syntactic diagonal lemma. This unification facilitates translating proofs among these theorems and deepens understanding of their interrelations in formal arithmetic theories extending a weakened form of Robinson’s arithmetic.