Foundation Universal Forcing Continuous Positive Ratio Arithmetic Invariant

The invariant arithmetic

The declaration continuous_positive_ratio_arithmetic_invariant is a formal statement inside the Recognition Science framework's machine-checked library of formal theorems. It concerns a particular way of building arithmetic from a recognition process, where recognition is a discrete record of events and their costs. The specific construction starts with a comparison operator on positive ratios, a rule that says how two positive numbers relate, and requires that this rule satisfy the framework's five laws of logic. From such a rule, the framework extracts an arithmetic object, a structure with a zero, a successor function, and the other ingredients of counting.

The declaration proves that this arithmetic object, built from the continuous positive-ratio comparison, is canonically equivalent to the arithmetic object extracted from any other admissible realization of the laws of logic. In plainer terms: no matter which valid comparison rule you start with, the counting structure you get is the same, up to a unique, structure-preserving correspondence. The proof works because every such extracted arithmetic object is an initial Peano algebra, a minimal model of the natural numbers, and any two initial Peano algebras are uniquely isomorphic. The declaration continuous_positive_ratio_arithmetic_invariant is a specific instance of a broader theorem, called Universal Forcing, which states that any two law-of-logic realizations force the same arithmetic surface.

The declaration does not claim that the continuous positive-ratio comparison is the only possible rule, nor that it is preferred over others. It does not claim that the framework derives the natural numbers from nothing; the Peano structure is a definitional choice within the framework, and the theorem states that this choice is invariant across realizations. It also does not claim anything about the empirical content of arithmetic, such as which physical systems instantiate it. The declaration is a statement about the internal consistency of the framework: its arithmetic is not an accident of one particular construction.

/-- The continuous positive-ratio realization has the same forced arithmetic
as every other realization. -/
noncomputable def continuous_positive_ratio_arithmetic_invariant
    (C : LogicAsFunctionalEquation.ComparisonOperator)
    (h : LogicAsFunctionalEquation.SatisfiesLawsOfLogic C)
    (S : LogicRealization.{0, 0}) :
    (arithmeticOf (LogicRealization.ofPositiveRatioComparison C h)).peano.carrier
      ≃ (arithmeticOf S).peano.carrier :=
  ArithmeticOf.equivOfInitial
    (arithmeticOf (LogicRealization.ofPositiveRatioComparison C h)) (arithmeticOf S)

/-- The Peano surface is available for the forced arithmetic of every
realization. -/
theorem peano_surface (R : LogicRealization) :
    ArithmeticOf.PeanoSurface (arithmeticOf R) :=
  ArithmeticOf.extracted_peanoSurface R

What this page does not claim

The declaration does not claim that the continuous positive-ratio comparison is the only possible rule or that it is preferred over others. It does not claim that the framework derives the natural numbers from nothing; the Peano structure is a definitional choice within the framework. It does not claim anything about the empirical content of arithmetic, such as which physical systems instantiate it.

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

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