Foundation Universal Forcing Strict Discrete Boolean Strict Boolean Arith Equiv

Two routes to the same arithmetic

In mathematics, a natural number is what you get when you start with zero and repeatedly add one. The Peano axioms capture this idea formally: they define a structure with a first element and a successor operation, and every structure satisfying them behaves like the familiar counting numbers. Different logical systems can build their own versions of this structure, and the question is whether they all agree.

The Recognition Science framework constructs arithmetic from a ledger, a discrete record of recognition events. One realization starts from Boolean propositions, where each event is either true or false, and the cost of moving between two states is 0 if they match and 1 if they differ. Another starts from positive ratios, comparing sizes. The declaration strictBoolean_arith_equiv_logicNat proves that the arithmetic forced by the Boolean realization is exactly the same as the framework's standard LogicNat structure. In plain terms, starting from two-valued logic and starting from the framework's canonical arithmetic construction lead to the same counting numbers.

A second declaration extends the point. It shows that positive ratios and Boolean propositions force the same arithmetic as well. The two results together form what the library calls the first strict cross-realization invariance theorem: different starting materials, the same forced outcome. The framework's cost function, which measures the price of recognition, is what drives this convergence.

What the declaration does not claim is that Boolean logic is the only way to build arithmetic, or that the framework's construction replaces the standard Peano axioms. It establishes an equivalence between two specific realizations within the framework. The result is a structural fact about the framework's own constructions, not a claim about all possible logical systems. It also does not say that true and false are the only possible states in every ledger; it says that when they are, the arithmetic that follows matches the canonical one.

/-- Strict Boolean forced arithmetic is canonically `LogicNat`. -/
def strictBoolean_arith_equiv_logicNat :
    (StrictLogicRealization.arith strictBooleanRealization).peano.carrier
      ≃ ArithmeticFromLogic.LogicNat :=
  (StrictLogicRealization.toLightweight strictBooleanRealization).orbitEquivLogicNat

/-- 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

Boolean logic is the only possible starting point for arithmetic. The framework's construction replaces the standard Peano axioms. True and false are the only possible states in every ledger.

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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