Gödeliin

Gödeliin is a term used in discussions of mathematical logic to denote a hypothetical formal system that embodies Gödelian self-reference through an internal fixed-point construction. Named after Kurt Gödel, the concept is used to discuss how a theory can express statements about its own proofs without asserting new, separate machinery outside the theory.

Formally, a Gödeliin system would extend a base arithmetic or similar formalism with a provability predicate and a fixed-point operator that yields a sentence that asserts its own unprovability within the system. The canonical Gödel sentence, constructed via arithmetization of syntax, is then internalized as part of the language itself. In this view, the self-referential sentence serves to illustrate the core incompleteness phenomenon: if the system is consistent, that sentence is true but unprovable within the system.

Usage and reach: The term is mainly encountered in pedagogical discussions, thought experiments, or speculative literature exploring stronger or alternative logics. It is not a standardized formalism with a canonical axiomatization, and different authors may offer different variants or naming conventions.

Significance: Gödeliin helps students and philosophers grasp the distinction between truth and provability and the way fixed points can encode meta-mundane statements about a theory.

See also: Gödel's incompleteness theorems; arithmetization of syntax; fixed-point lemma; self-reference.