IndisputableMonolith.Foundation.UniversalForcing.Strict.CategoricalMathlib
This module implements the LogicNat natural numbers object recursor via Lean's inductive pattern matching on zero and successor. It supplies the concrete carrier for the strict categorical NNO in the UniversalForcing framework. Category theorists bridging Lawvere-style arithmetic to Lean foundations cite it for the universal recursion property. The module consists of direct definitions establishing the recursor reductions and uniqueness without extra axioms.
claimThe recursor $R$ for the natural numbers object $N$ satisfies $R(z,s,0)=z$ and $R(z,s,S(n))=s(n,R(z,s,n))$, where $z$ is the base value and $s$ the successor step, realizing the universal property of primitive recursion on $N$.
background
The strict categorical realization in UniversalForcing.Strict uses LogicNat as the canonical carrier for the natural numbers object (NNO) in a Lawvere-style setting. This module builds directly on the upstream Categorical hook, which positions LogicNat as the NNO surface from CategoricalLogicRealization and notes that future work may refine it to Mathlib's full category-theory NNO API. It introduces the recursor together with its zero and successor reduction rules plus the universal existence and uniqueness statements for the NNO.
proof idea
This is a definition module, no proofs. It defines the recursor by pattern matching on the inductive LogicNat type, states the zero and successor cases explicitly, and records the universal existence and uniqueness properties as direct consequences of the inductive structure.
why it matters in Recognition Science
This module supplies the recursor universal property that the downstream MathlibNNO module exposes to Mathlib's CategoryTheory namespace. It completes the concrete implementation of the strict categorical realization hook, grounding the arithmetic primitives required by the T0-T8 forcing chain. The module thereby allows the Recognition framework to interface categorical NNOs with Lean's native induction while keeping theorem content in the already-proved CategoricalMathlib layer.
scope and limits
- Does not provide a full Mathlib CategoryTheory NNO API beyond the recursor.
- Does not treat non-strict or probabilistic variants of the forcing.
- Does not address higher categorical constructions such as limits or exponentials.
- Does not prove equivalence with other arithmetic encodings outside LogicNat.