Foundation Arithmetic From Logic Log Generator Ne Zero

The generator's nonzero step

The natural numbers, 0, 1, 2, 3 and so on, are usually introduced as a starting point for arithmetic. The Recognition Science framework instead derives them from a more primitive structure. Its machine-checked library of formal theorems shows that a non-trivial comparison operator, one that distinguishes some values from others, forces an identity element and a generator, a positive real number different from 1. Repeated multiplication by that generator, together with the identity, produces the sequence {1, γ, γ², γ³, ...}, which has exactly two ways to build a new element: be at the identity, or take one more step. That two-constructor structure is the natural-number structure.

The declaration log_generator_ne_zero establishes a small but essential fact about this generator: its logarithm is not zero. In plain terms, the step that moves from one number to the next is genuinely a step, not a standstill. If the logarithm of the generator were zero, the generator itself would equal 1, and the whole orbit would collapse to a single point. The declaration rules out that collapse, ensuring that the counting process actually advances. This is a theorem in the framework's library, proved from the axioms of the system, not an assumption added by hand.

The construction is deliberately independent of any base for writing numbers. Nothing in it references base 10, base 2, or any positional system. The only primitives are the identity element and the step operation. The natural numbers as forced by the Law of Logic form an inductive type with two constructors, identity and step, mirroring the orbit {1, γ, γ², γ³, ...} as the smallest subset of positive reals closed under multiplication by γ and containing 1.

What the declaration does not claim is just as important. It does not assert that the generator has any particular numerical value, such as the golden ratio, nor does it say which real number the step corresponds to. That value is fixed by further results in the framework, not by this declaration alone. The declaration also does not claim that the natural numbers so constructed are the same as the usual ones in every respect; it establishes the structural shape, with addition, multiplication, and order all defined on this new type, and a translation to the standard natural numbers exists but is a separate theorem.

/-- **Peano P1, contrapositive**: every successor differs from zero. -/
theorem succ_ne_zero (n : LogicNat) : succ n ≠ zero := by
  intro h; cases h

/-- The natural numbers as forced by the Law of Logic.

`identity` represents the zero-cost element (the multiplicative
identity in the orbit). `step` represents one more iteration of the
generator. The two-constructor structure mirrors the orbit
{1, γ, γ², γ³, ...} as the smallest subset of ℝ₊ closed under
multiplication by γ and containing 1. -/
inductive LogicNat : Type
  | identity : LogicNat
  | step     : LogicNat → LogicNat
  deriving DecidableEq, Repr

/-- The natural numbers as forced by the Law of Logic.

`identity` represents the zero-cost element (the multiplicative
identity in the orbit). `step` represents one more iteration of the
generator. The two-constructor structure mirrors the orbit
{1, γ, γ², γ³, ...} as the smallest subset of ℝ₊ closed under
multiplication by γ and containing 1. -/
inductive LogicNat : Type
  | identity : LogicNat
  | step     : LogicNat → LogicNat
  deriving DecidableEq, Repr

What this page does not claim

The declaration does not assign a specific numerical value to the generator. The declaration does not claim the constructed natural numbers are identical to standard natural numbers in every property. The declaration does not by itself establish the golden ratio or any other particular constant as the generator.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/ArithmeticFromLogic.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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