Foundation Primitive Recognition Calculus Rigidity Base Initiality Is Peano Mode

The Peano model condition

The natural numbers are the familiar counting sequence 0, 1, 2, 3, and so on. The Italian mathematician Giuseppe Peano captured their essential behavior in three famous axioms: zero is not the successor of any number, every number has a distinct successor, and mathematical induction works. Any structure satisfying these behaves exactly like the natural numbers.

The declaration IsPeanoModel packages those same three requirements as a single formal condition on a more abstract object called a delta-algebra, a carrier set with a distinguished starting point and a distinction step. The declaration is a definition, a choice of what counts as a Peano model, not a theorem about the world.

In Recognition Science, the framework models the act of distinction as a delta-algebra: a collection of objects, a first object, and a way to step from one object to the next. The framework's machine-checked library of formal theorems proves that any delta-algebra satisfying the Peano condition is rigid: there is exactly one structure-preserving map from the standard counting model to it, and that map is a bijection. The library's theorem base_rigidity bundles this initiality, categoricity, and uniqueness into one statement.

What the declaration does not claim is that the physical universe is literally a Peano model. It establishes a uniqueness result within the framework: given the axioms, the counting structure is forced. Whether real recognition events form such a structure is a separate question, left open by the declaration itself.

/-- A δ-algebra is a **Peano model** when it is a non-degenerate, minimal
realization of distinction: the step is injective, the base point is never a step,
and the carrier is generated by the base point under the step (induction /
minimality). These are the properties a candidate carrier must have to count as
"the same object as the δ-base," rather than a strictly larger or degenerate
structure. -/
structure IsPeanoModel (A : DeltaAlgebra) : Prop where
  succ_injective : Function.Injective A.succ
  zero_not_succ : ∀ x, A.succ x ≠ A.zero
  induction : ∀ (P : A.carrier → Prop),
    P A.zero → (∀ x, P x → P (A.succ x)) → ∀ x, P x

/-- The full base-rigidity conjunction: initiality, categoricity, and uniqueness
of the isomorphism. The δ-base is a canonical object — forced up to unique
isomorphism. THEOREM (choice-free; see the audits below). -/
theorem base_rigidity :
    target_initial ∧ target_categorical ∧ target_unique_iso :=
  ⟨base_initial, base_categorical, base_unique_iso⟩

theorem base_unique_iso : target_unique_iso := by
  intro M h
  refine ⟨{ default := ⟨baseHom M, baseRec_injective M h, baseRec_surjective M h⟩,
            uniq := ?_ }⟩
  intro f
  apply Subtype.ext
  exact (baseInitial M).uniq f.1

What this page does not claim

The physical universe is a Peano model. The declaration proves the Peano axioms are true of reality. The rigidity theorem applies to structures that do not satisfy the Peano condition.

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/PrimitiveRecognitionCalculus/Rigidity/BaseInitiality.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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