Gödelian Ternary Logic (GTL): A Logic Embedding Incompleteness

This paper introduces Gödelian Ternary Logic (GTL), a three-valued system where the third truth value D (Dao) formally embodies Gödel’s first incompleteness theorem. Unlike traditional ternary logics where the third value denotes ”unknown” or ”undefined”, D intrinsically represents statements that are true in the standard model but unprovable within the system. The logic features a non-classical negation (¬D = D), provability-centric implication, and controlled paraconsistency. GTL’s distinctive power lies in its capacity to: (1) formalize system boundaries through truth values, (2) resolve logical paradoxes while preserving consistency for decidable statements, (3) model unprovable implications, and (4) enable formal systems to reason about their own limitations. Truth tables and model theory demonstrate how D propagates through operations while preserving core metamathematical properties.