This paper reframes Gödel’s incompleteness phenomena as part of a broader structure of irreversible information. It argues that every act of resolving uncertainty—whether in logic, scientific inquiry, or cognition—requires erasing alternative possibilities, and that erasure carries an objective thermodynamic cost. This “informational residue” connects epistemic collapse to entropy production and provides a conceptual bridge toward an emerging framework of informational thermophysics, where irreversible information becomes physically consequential. This work serves as the philosophical and informational foundation for later developments linking informational cost to thermodynamic irreversibility and, ultimately, to geometric curvature through the relation R = \alpha C^2.