arith_universal_initial'
Every law-of-logic realization induces an arithmetic structure on its Peano carrier that is canonically equivalent to the reference natural numbers built from the logic orbit. Researchers in universal forcing within Recognition Science cite this to establish that arithmetic extraction is independent of the specific realization chosen. The definition is a direct one-line wrapper that applies the orbit equivalence supplied by the realization.
claimFor any law-of-logic realization $R$, the carrier of the Peano arithmetic induced by $R$ is equivalent to the natural numbers forced by the law of logic (denoted LogicNat).
background
A law-of-logic realization supplies a carrier set, a comparison cost, a zero-cost identity, and a generator step whose orbit generates the arithmetic. LogicNat is the inductive type with constructors identity (the multiplicative identity 1) and step (iterating the generator), forming the smallest subset of positive reals closed under multiplication by the generator and containing 1. The module states that every such realization carries canonically equivalent forced arithmetic. Upstream, LogicNat mirrors the orbit {1, γ, γ², ...} and LogicRealization is the lean structure carrying the identity/step data plus structural laws.
proof idea
One-line wrapper that applies the orbit equivalence supplied directly by the realization.
why it matters in Recognition Science
This definition supplies the initial equivalence step of the universal forcing meta-theorem, guaranteeing that forced arithmetic is the same for any realization. It supports the module claim that every law-of-logic realization carries canonically equivalent forced arithmetic and aligns with the Recognition Science program of extracting invariant arithmetic from the law of logic. No open questions are addressed here.
scope and limits
- Does not exhibit the explicit form of the equivalence map.
- Does not incorporate physical constants or specific realizations.
- Does not preserve additional structure such as order or topology.
- Does not address decidability or computational properties.
formal statement (Lean)
21noncomputable def arith_universal_initial' (R : LogicRealization) :
22 (arithmeticOf R).peano.carrier ≃ ArithmeticFromLogic.LogicNat :=
proof body
Definition body.
23 R.orbitEquivLogicNat
24
25/-- **Universal Forcing Meta-Theorem.**
26
27For any two Law-of-Logic realizations, the forced arithmetic objects are
28canonically equivalent. -/