Tarski's undefinability theorem [SEP] named after
Tarski's undefinability theorem
Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1936, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics. Informally, the theorem states that "arithmetical truth cannot be defined in arithmetic".
The theorem applies more generally to any sufficiently strong formal system, showing that truth in the standard model of the system cannot be defined within the system.
History.
In 1931, Kurt Gödel published the incompleteness theorems,
. They were followed by Tarski's undefinability theorem on the formal undefinability of truth, Church's proof that Hilbert's Entscheidungsproblem is unsolvable, and Turing's theorem that there is no algorithm to solve the halting problem.
Formal systems: completeness, consistency, and effective axiomatization.
The incompleteness theorems apply to formal systems that are of sufficient complexity to express the basic arithmetic of the natural numbers and which are consistent, and effectively axiomatized, these concepts being detailed below. Particularly in the context of first-order logic,