IndisputableMonolith.Foundation.UniversalForcing.CategoricalRealization
This module supplies the canonical categorical realization of the universal forcing framework by means of the LogicNat Peano object. Researchers establishing links between categorical logic and forced arithmetic in Recognition Science would cite it. The module packages the natural-number object from the upstream CategoricalLogicRealization in initial-Peano-algebra language without new category theory.
claimThe canonical categorical realization is given by the Peano object drawn from logic natural numbers, yielding the equivalence between categorical arithmetic structures and the initial Peano algebra.
background
The module resides in the Foundation.UniversalForcing layer and imports CategoricalLogicRealization. That upstream module supplies a Lawvere-style realization hook for the Universal Forcing program, packaging the natural-number object idea in the initial-Peano-algebra language used by ArithmeticOf. The present module specializes the hook to a canonical form via the LogicNat Peano object.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
This module feeds AxiomAudit, which supplies the reproducible theorem surface for the Universal Forcing Lean program, and Invariance.Universal, whose general theorem states that every Law-of-Logic realization carries canonically equivalent forced arithmetic. It supplies the categorical bridge required for the universal forcing equivalence.
scope and limits
- Does not rebuild category theory from scratch.
- Does not address realizations outside the initial Peano algebra.
- Does not prove the downstream equivalence theorems.