Knowra Tarski's undefinability theorem Tarski's undefinability theorem Tarski's undefinability theorem says that a sufficiently expressive, consistent formal theory of arithmetic cannot define within itself a predicate that correctly captures truth for every sentence in its own language.
Diagonal lemma : A result that lets a formal language construct a sentence that refers, via coding, to its own code. It builds the self-referential sentence that exposes the limits of a proposed truth definition.
First-order arithmetic : The formal study of natural numbers using first-order logic and arithmetic axioms. It is the setting in which the theorem rules out a definable predicate for full sentence truth.
Alfred Tarski : A Polish-American logician and mathematician known for work on truth, models, and formal theories. He proved the undefinability result and developed a semantic account of truth.
Gödel's incompleteness theorems : Theorems showing that sufficiently strong consistent formal systems leave some arithmetic truths unproved and cannot prove their own consistency. Both results use arithmetized syntax and self-reference, though they establish different limitations.
Gödel numbering : A method that assigns natural numbers to symbols, formulas, and proofs in a formal system. It lets arithmetic talk about sentences and predicates in the language whose truth is at issue.
Formal language : A set of strings formed from symbols according to explicit rules. The theorem concerns truth for sentences in a specified formal language, not truth in every possible language.
The Concept of Truth in Formalized Languages : Alfred Tarski's work establishing a formal semantic definition of truth for formalized languages. It presents the distinction between a language and a stronger metalanguage that underlies the theorem.
Truth predicate : A predicate intended to apply to sentences that are true in a specified language or interpretation. The theorem's central negative result concerns defining such a predicate within the same arithmetic language.
T-schema : A schema linking a sentence's truth to the sentence itself, as in “'Snow is white' is true if and only if snow is white.” A genuine truth predicate must satisfy corresponding instances, which the internal predicate cannot do for all sentences.
Satisfaction relation : A relation that states whether a formula is true under an assignment of values to its free variables. Truth for sentences is the closed-formula case of satisfaction, which can be defined externally.
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