Tarski Undefinability of Truth

Origin. Tarski (1933, 1936). Arithmetical truth is not arithmetically definable. The semantic counterpart of Gödel incompleteness and the formal resolution of the Liar paradox.

Models. No formula in the language of arithmetic defines the set of Gödel numbers of true sentences. A language cannot contain its own truth predicate on pain of contradiction.

Formalism.

Diagonal lemma: For any φ(x) there is a sentence ψ with ⊢ ψ ↔ φ(⌜ψ⌝). Self-reference via arithmetization.

Undefinability theorem: No formula True(x) satisfies N ⊨ True(⌜φ⌝) ↔ φ for all sentences φ.

Proof: Suppose True(x) exists. Diagonalize on ¬True(x): get ψ ↔ ¬True(⌜ψ⌝). Then ψ is true iff not true — contradiction.

Object vs metalanguage: Truth for a language is definable only in a stronger metalanguage. Tarski hierarchy of languages.

Partial truth predicates: Truth for Σ_n sentences is Σ_n-definable. Full truth is not definable at any fixed level.

T-schema: True(⌜φ⌝) ↔ φ, as a material adequacy condition.

Symbols.

Symbol	Unicode	Name	Meaning
True(x)	—	Truth predicate	Alleged definition of truth
⌜φ⌝	—	Gödel number	Code of sentence φ
N	—	Standard model	Natural numbers
↔	U+2194	Biconditional	Material equivalence

Metatheory. Contrast with provability, which is arithmetically definable (Σ⁰₁) — this gap between definable provability and undefinable truth is exactly what drives incompleteness. Truth strictly transcends the object language; stratification is the price of consistency.

Applies to. Theory of truth. Semantic paradoxes. Foundations of model theory. Reflection principles and hierarchies.

Limitations. Requires arithmetization of syntax. Language-relative: truth exists, just not internally. Stratified truth predicates are cumbersome; revision and fixed-point theories offer alternatives.