IndisputableMonolith.Foundation.UniversalForcing.CategoricalRealization
The CategoricalRealization module supplies the canonical categorical realization of arithmetic via the LogicNat Peano object inside the Universal Forcing program. Researchers extending Lawvere-style natural-number objects to forced arithmetic would cite it when linking initial Peano algebras to categorical realizations. The module consists of two sibling definitions that import and adapt the CategoricalLogicRealization hook without new category-theoretic machinery.
claimThe canonical categorical realization is the object $R$ obtained from the LogicNat Peano algebra such that categorical arithmetic is equivalent to the initial Peano structure, written $R : \mathsf{LogicNat} \simeq \mathsf{PeanoAlgebra}$.
background
The module sits inside the Universal Forcing program and imports CategoricalLogicRealization, whose doc states it packages the natural-number object idea in the same initial-Peano-algebra language used by ArithmeticOf. It does not rebuild category theory; it supplies the canonical hook via the LogicNat Peano object. The two sibling declarations are categoricalRealization and categorical_arith_equiv_logicNat.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module feeds the AxiomAudit surface for reproducible theorems and supplies the realization used by Invariance.Universal, whose doc states that every Law-of-Logic realization carries canonically equivalent forced arithmetic. It therefore closes the categorical-to-arithmetic link required by the Universal Forcing chain.
scope and limits
- Does not rebuild any category theory from axioms.
- Does not contain theorem statements or proofs.
- Does not address forcing chains beyond the realization hook.
- Limits scope to packaging the LogicNat Peano object.