VR-Numbers: Operational Superstructures of Integers, Rationals, Reals, and Complex Numbers over the Natural Numbers of VR

Building on VR (Reznik, 2026, DOI: 10.5281/zenodo.20212092), a formal axiomatization ofarithmetic with three primitives {∅, →, t} and four axioms, we develop VR-Numbers — aprogram of operational superstructures yielding ℤ, ℚ, ℝ, and ℂ over VR's naturalnumbers ℕ. The construction uses no ordered pairs as ontological objects; eachnumerical extension is defined as a formal language of expressions with equivalencerelations and operations. The ontological foundation of the entire system is the singleprimitive ∅; all numerical objects — including ℕ — arise as operational constructions ofprogressively greater depth. The classical actual infinity is replaced by operationalinfinity: A4 (induction) is read as an operational principle, not as a postulate ofcompleted infinite totalities. The two-dimensionality of ℂ is derived from the dualitypresent in axiom A1 of VR (the two generating implications F → F and F → ⊤), rather thanpostulated through pairs of reals. ℤ_VR, ℚ_VR, and ℂ_VR are shown to be isomorphic tothe corresponding standard structures; ℝ_VR is isomorphic to the field of computablereal numbers (a countable subfield of classical ℝ), reflecting the operational character ofthe construction.