Tarski's Undefinability Theorem

Tarski's undefinability theorem (1933) establishes that for any sufficiently expressive formal language L, the concept of truth for L cannot be defined within L itself. The proof constructs a self-referential sentence analogous to the liar paradox: if truth were definable, one could formulate a sentence asserting its own falsehood, producing a contradiction. The theorem is the semantic counterpart to Gödel's syntactic incompleteness: where Gödel shows that some truths are unprovable, Tarski shows that truth itself cannot be captured by the system's own vocabulary. The consequence is that semantics must always be pursued from a meta-language — there is no semantic closure within a single formal frame. This has implications not only for logic but for any system that attempts to represent its own representational adequacy, from cognitive science to the theory of meaning.

Tarski's theorem is not a limitation of particular languages but a structural law: no language can be its own metalanguage. The boundary between object and meta-level is not a convention but a necessity, and every attempt to erase it regenerates the liar.