bool_peano_surface
plain-language theorem explainer
The Boolean realization forces an arithmetic object whose Peano surface obeys zero distinct from successors, successor injectivity, and the induction schema. Discrete-model researchers in Recognition Science cite this as the first non-continuous verification that universal forcing yields standard arithmetic. The proof is a direct one-line application of the general peano_surface theorem to boolRealization.
Claim. Let $A$ be the arithmetic object extracted from the Boolean realization (carrier Bool, cost Nat). Then $A$ satisfies: $A$.peano.zero $neq$ $A$.peano.step$(x)$ for all $x$, the successor map is injective, and for every predicate $P$ on the carrier, $P$(zero) and $forall x, P(x) to P$(step $x)$ imply $forall x, P(x)$.
background
The DiscreteLogicRealization module supplies the second Law-of-Logic realization: a discrete Boolean carrier that serves as the first non-continuous test case for Universal Forcing. boolRealization defines Carrier as Bool and Cost as Nat, with zero as false and comparison via boolCost. ArithmeticOf packages the Peano object forced by any realization together with its initiality. PeanoSurface is the Prop asserting the three surface axioms: zero differs from every successor, successor is injective, and induction holds for all predicates on the carrier. The upstream theorem peano_surface states that for every realization R the extracted arithmeticOf R satisfies PeanoSurface.
proof idea
The proof is a one-line wrapper that applies the general theorem UniversalForcing.peano_surface to boolRealization.
why it matters
This theorem verifies that the discrete Boolean realization inherits the Peano surface from the universal forcing construction, closing the first non-continuous test case in the DiscreteLogicRealization module. It confirms that the forcing chain produces standard arithmetic even for propositional carriers. No downstream uses are recorded yet.
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