Tarski’s Undefinability Theorem

The simplest summary of Tarski’s deepest result is that truth, for any sufficiently expressive system, is always partly outside the system’s own reach. The proof turns on the oldest paradox in philosophy: the Liar, the sentence that says of itself that it is not true. If a sufficiently rich language contained a complete and consistent predicate for its own truth, you could use a diagonal construction—the same technique Gödel used to build a sentence asserting its own unprovability—to construct a Liar sentence inside the language. Since that sentence produces contradiction, and the systems we care about are not contradictory, the predicate cannot exist. Truth, for a language expressive enough to matter, must be defined in a richer metalanguage above—one the object language cannot reach, preventing the Liar from reassembling. The result generalizes from formal languages to a structural expectation for any sufficiently expressive symbolic process: the self-evaluating system, the model asked