Self-reference

Self-reference occurs when a statement, expression, or system refers to itself. It can produce fixed points, semantic paradoxes, and limits on formal proof.

Diagonal lemma: A theorem in logic that lets a sentence in a formal theory assert a property of its own Gödel number. It constructs self-referential sentences inside arithmetic without placing a sentence literally inside itself.
Liar paradox: The paradox generated by a sentence that declares itself false. Its truth evaluation reverses itself because the sentence refers to its own truth status.
Formal system: A collection of symbols, formation rules, axioms, and inference rules used to derive statements. Self-reference in mathematics depends on what a formal system can express and prove.
Circular reasoning: An argument that uses its conclusion, directly or indirectly, as support for that conclusion. Circular arguments loop through premises, while self-reference need not be an argument or a fallacy.
Gödel numbering: A method that assigns natural numbers to symbols, formulas, and proofs in a formal system. Encoding formulas as numbers lets arithmetic represent statements about its own syntax.
Grelling–Nelson paradox: A paradox arising from asking whether the adjective “heterological” applies to itself. It applies self-reference to the classification of words rather than to truth alone.
Truth predicate: A predicate that assigns truth or falsity to sentences in a language. Self-referential truth claims become paradoxical when a language can apply its truth predicate to its own sentences.
Autology: A word that describes a property it itself possesses, such as “short” when the word is short. Autological words apply to themselves but do not necessarily refer to themselves.
Fixed point (mathematics): An input that a function maps to itself. Self-referential constructions often seek a sentence unchanged by a transformation of its description.
Russell's paradox: The contradiction obtained by considering the set of all sets that do not contain themselves. Membership applied to a set defined by its own membership condition produces contradiction.

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