Tight-Apartness Empty Sets in Paraconsistent Constructive Set Theory: A Class of Absolute Nothings Beyond Classical Emptiness

This paper develops a fully explicit paraconsistent constructive framework in which “absolute nothing” can be rigorously formalized without collapse into trivial inconsistency. Working over Constructive Zermelo–Fraenkel set theory with the Regular Extension Axiom (CZF + REA), evaluated in LP-based and bilattice-valued semantics, we define tight-apartness empty (TAE) objects as sets that are apart from every object, including themselves. We provide a complete axiomatization of equality for glut-valued objects, construct explicit four-valued bilattice models validating CZF + REA at designated values, and prove a relative consistency theorem for PCZF + ∃N TAE(N) relative to CZF + REA + Inacc. We further establish an exact duality between equality-paraconsistent semantics and TAE objects: every non-trivial equality-paraconsistent model admits a canonical TAE normal form, and conversely every model with a TAE object is necessarily equality-paraconsistent. The results isolate contradiction at the level of identity while preserving full classical and constructive behavior for ordinary mathematical objects. Tight-apartness empty sets thereby provide a predicative, constructively grounded notion of absolute boundary objects that resist all positive characterization, fail extensionality, admit non-unique instantiation under large-set assumptions, and collapse under positive attribution.