Difference and Homogeneity: The Ontological Primitive. Co-implication with Time, Possibility and Multiplicity: A Formal Proof of Undefinability

This paper demonstrates that difference (Diff) is an ontological primitive in a technically precise sense: it cannot be defined within any sufficiently expressive formal system without already presupposing it, and its negation - complete homogeneity (Hom) - leads to performative contradiction in any system that affirms it. The proof extends to the entire linguistic hierarchy: no metalanguage can capture (Diff) without presupposing it, distinguishing this result from Tarski's undefinability of truth. Time, multiplicity, and possibility share the same character: none can be defined without presupposing the others or presupposing difference. The four terms form a system of reciprocal co-implication - not a derivational hierarchy - whose total negation is contradictory in any possible world accessible to any operating system. The privileged entry role of (Diff) in the exposition is purely methodological: all four terms are ontologically co-primary. Demonstrations are carried out in FOL extended with S4 modal operators, with an internally defined accessibility relation: a world is accessible if (Diff) can be bordered in it. We justify this system choice, note that the proof is itself an instance of the constitutive circularity it describes, and address foreseeable objections. The results bear directly on the foundations of epistemology, logic, and any discipline that uses formal systems as its substrate.