Tarski's Undefinability Theorem and Truth

Core Idea

Tarski's undefinability theorem shows that the set of true sentences of a model cannot be defined within the language of that model itself. Even though satisfaction can be defined in meta-mathematics, there is no formula in the object language expressing truth in the model. This fundamental limitation constrains what logical definability can achieve.

Explainer

From your study of model interpretation and satisfaction, you know that truth in a model is defined at the *metalevel*: we say a formula φ is satisfied by structure M (written M ⊨ φ) by an inductive clause-by-clause definition given *outside* the object language. But could we bring this definition *inside*? Could we write a formula Truth(x) in the language of arithmetic, say, such that Truth(⌜φ⌝) is true if and only if the sentence φ is true? Tarski's undefinability theorem proves this is impossible.

The argument is a logical version of the Liar paradox. Begin by assuming, for contradiction, that there exists a formula Truth(x) in the language of arithmetic that correctly identifies the Gödel codes of true sentences. Using the diagonal lemma (a consequence of the expressibility of syntax within arithmetic), construct a sentence L such that L is provably equivalent to ¬Truth(⌜L⌝) — a sentence that "says" it is not true. Now ask whether L is true: if L is true, then Truth(⌜L⌝) holds, but then ¬Truth(⌜L⌝) is false, contradicting L's truth. If L is false, then ¬Truth(⌜L⌝) holds, but L is equivalent to ¬Truth(⌜L⌝), so L is true — a contradiction either way. Therefore no such Truth(x) formula can exist.

The key distinction Tarski's theorem establishes is between *object language* and *metalanguage*. Satisfaction (and truth in a model) *can* be defined — but only from outside the language, in a richer metalanguage that can refer to the original language's formulas as objects. This is not a defect of any particular formalization; it is a fundamental semantic limitation. The Tarski hierarchy captures this: truth for level-n sentences can be defined in a level-(n+1) metalanguage, but this generates an infinite regress rather than a single unified truth predicate. No language can pull itself up by its own semantic bootstraps.

The contrast with what *is* definable is illuminating. From your prerequisite work on Beth definability, you know that many semantic notions (satisfaction, definability, isomorphism) are perfectly well-defined in the metalanguage — they just cannot be *expressed* inside the object language as a formula. Arithmetic can talk about its own *syntax* (via Gödel coding) but not about its own *semantics* (what that syntax means). This is precisely the gap that Gödel's incompleteness theorems also exploit: the coding of syntax allows self-reference, but truth cannot be captured, only provability — and provability and truth diverge for sufficiently rich theories.