VR-Numbers: Operational Superstructures of Integers, Rationals, Reals, and Complex Numbers over the Natural Numbers of VR
Vitaly Reznik · Zenodo (CERN European Organization for Nuclear Research) · 2026
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 natural numbers ℕ . Theconstruction uses no ordered pairs as ontological objects; each numerical extension is defined as aformal language of expressions with equivalence relations and operations. The ontologicalfoundation of the entire system is the single primitive ∅; all numerical objects — including ℕ —arise as operational constructions of progressively greater depth. The classical actual infinity isreplaced by operational infinity: A4 (induction) is read as an operational principle, not as apostulate of completed infinite totalities. The two-dimensionality of ℂ is structurally grounded inthe duality present in axiom A1 of VR (the two generating implications F → F and F → ⊤), ratherthan postulated through pairs of reals; the algebraic coupling of the two axes requires an additionaljoining axiom (i² = ℤ 1). ℚ_VR, ℂ _VR, and ℂ_VR are shown to be isomorphic to thecorresponding standard structures; ℝ _VR is isomorphic to the field of computable real numbers —a countable subfield of classical ℝ — reflecting the operational character of the construction.Version 1.0.2 (2026) adds Part VIII collecting methodological observations from the Lean 4 formalisation of Parts II–V (companion publication, DOI 10.5281/zenodo.20352057). No axioms, definitions, or theorems of Parts I–VII are altered.