Foundation Universal Forcing Strict Discrete Boolean Strict Boolean Realization

The Boolean realization

In classical logic, a proposition is either true or false. Recognition Science asks what arithmetic arises when the cost of recognition is measured on just these two values. The recognition cost, the price of telling two states apart, is 0 when the states match and 1 when they differ. This is the simplest possible discrete ledger: a record with two entries and a forced difference between them.

The definition strictBooleanRealization builds a full logical structure on this two-value carrier. Its arithmetic, the counting system that emerges from iterating the native generator, is canonically equivalent to LogicNat, the framework's natural numbers built from logic. The key equivalence is proved: the arithmetic of the Boolean realization is isomorphic to that of the positive ratios, meaning both force the same counting structure. This is a theorem in the machine-checked library, with the isomorphism established by strictPositiveRatio_arith_equiv_strictBoolean.

The carrier orbit is periodic, cycling through true and false, but the strict forced arithmetic is not the finite image inside Bool. It is the free iteration object derived from the native generator. This distinction matters: the finite two-value system, when forced to count, produces the infinite natural numbers, not a two-element arithmetic. The Boolean realization is a strict, discrete case of the framework's universal forcing, showing that even the most minimal logical substrate generates the same arithmetic as the continuous positive ratios.

What this does not claim is that the Boolean system itself is the natural numbers, or that it contains the full structure of the positive ratios. The equivalence is at the level of the forced arithmetic, the counting structure, not the carriers themselves. The Boolean realization is one model among many, and its value is in demonstrating invariance: different starting points, from continuous ratios to discrete truth values, converge on the same arithmetic.

/-- 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

The Boolean carrier itself is not the natural numbers; only its forced arithmetic is equivalent. The Boolean realization does not contain the full structure of the positive ratios, only the same arithmetic. This result does not prove that the Boolean system is the unique minimal realization; other strict realizations may exist.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: