A formal result that no sufficiently expressive formal language (for example, one that can represent basic arithmetic) can contain a formula that correctly and uniformly defines the truth predicate for sentences of that same language.

Principle

When a language is powerful enough to encode its own syntax and perform basic arithmetic, any attempt to define 'Truth' for that language from within it runs into self-referential constructions that prevent a consistent, total truth-defining formula.

Demonstration

In first-order arithmetic one shows that if a formula True(x) were to hold exactly for Gödel numbers of true sentences, the diagonal lemma yields a sentence that asserts its own falsehood relative to True, producing contradiction or failure of correctness of True(x).

Misapplication

Assuming Tarski's theorem forbids every internal talk of truth and therefore banning any metalanguage discussion; or claiming it applies to weak languages (like propositional calculus) that cannot represent the required syntactic coding.

Consequence

One consequence is the need to treat truth as a concept defined in a richer metalanguage or to use stratified truth predicates or hierarchies of languages rather than a single uniform internal truth predicate.

Reversal

If a language could internally define its own full truth predicate without contradiction, many standard incompleteness and undefinability arguments would fail, collapsing the separation between object language and metalanguage.

Boundary

Applies to languages capable of arithmetization and sufficient self-reference (e.g., Peano arithmetic). It does not apply to finite propositional languages or fragments that cannot represent syntactic Gödel-numbering.

Semantic Tension

Tension exists between the intuitive semantic notion of truth (a global property of sentences) and the formal notion of definability inside a language; truth resists internal specification while provability may be internally capturable.

Synthesis

Tarski's theorem identifies a fundamental boundary: languages that can express their own syntax cannot internally provide a complete, correct truth predicate, forcing either a metalanguage shift or restricted truth notions.