Foundation Universal Forcing Strict Discrete Boolean Strict Positive Ratio Arith

Two routes to one arithmetic

In ordinary mathematics, a positive ratio is a quantity like 3/2, and a Boolean value is a truth state like true or false. These seem unrelated. The declaration strictPositiveRatio_arith_equiv_strictBoolean establishes that, inside the Recognition Science framework, the arithmetic forced by comparing positive ratios is exactly the arithmetic forced by Boolean logic. More precisely, the carrier sets of the two forced arithmetic structures are in bijection, meaning each element of one corresponds to exactly one element of the other, and the arithmetic operations align through that correspondence.

The framework models recognition, a discrete record of events, with a forced cost. The declaration shows that two different starting points, one based on ratios and one based on truth values, converge on the same arithmetic structure. This is a theorem in the framework's machine-checked library of formal theorems, proved by composing two existing equivalences: the first links the ratio-based arithmetic to a canonical natural-number structure, and the second links the Boolean-based arithmetic to the same structure. The proof relies on the fact that both are initial objects in the relevant category, so they must be isomorphic.

The declaration does not claim that ratios and Booleans are the same thing, nor that one reduces to the other. It claims only that, under the framework's specific construction, the two forced arithmetic structures coincide. It also does not claim that this equivalence holds outside the framework's axioms, nor that it extends to other logical or arithmetic systems. The result is internal to the Recognition Science framework, a structural invariance across two of its realizations.

What this changes: the framework gains a robustness check. If two very different starting points, continuous ratios and discrete truth values, force the same arithmetic, then that arithmetic is not an artifact of one particular choice. The declaration strengthens the case that the framework's arithmetic is canonical, at least within its own terms.

/-- First strict cross-realization invariance theorem:
positive ratios and Boolean propositions force the same arithmetic. -/
noncomputable def strictPositiveRatio_arith_equiv_strictBoolean
    (C : ComparisonOperator) (h : SatisfiesLawsOfLogic C) :
    (StrictLogicRealization.arith (PositiveRatio.strictPositiveRatioRealization C h)).peano.carrier
      ≃ (StrictLogicRealization.arith strictBooleanRealization).peano.carrier :=
  ArithmeticOf.equivOfInitial
    (StrictLogicRealization.arith (PositiveRatio.strictPositiveRatioRealization C h))
    (StrictLogicRealization.arith strictBooleanRealization)

/-- First strict cross-realization invariance theorem:
positive ratios and Boolean propositions force the same arithmetic. -/
noncomputable def strictPositiveRatio_arith_equiv_strictBoolean
    (C : ComparisonOperator) (h : SatisfiesLawsOfLogic C) :
    (StrictLogicRealization.arith (PositiveRatio.strictPositiveRatioRealization C h)).peano.carrier
      ≃ (StrictLogicRealization.arith strictBooleanRealization).peano.carrier :=
  ArithmeticOf.equivOfInitial
    (StrictLogicRealization.arith (PositiveRatio.strictPositiveRatioRealization C h))
    (StrictLogicRealization.arith strictBooleanRealization)

What this page does not claim

The declaration does not claim that ratios and Booleans are the same thing. It does not claim the equivalence holds outside the framework's axioms. It does not claim the result extends to other logical or arithmetic systems.

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

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