Foundation Primitive Recognition Calculus Rigidity Base Initiality Hom Eq Base R

The uniqueness of counting

Counting is the first act of distinction: a starting point, then a step to the next thing. The natural numbers are the standard model of this act, with zero and the successor function. The declaration hom_eq_baseRec is a theorem in the framework's machine-checked library of formal theorems. It states that any structure that respects this basic counting pattern must be built in exactly one way: by starting at the distinguished point and repeatedly applying the distinction step.

To state this precisely, the framework defines a ledger, a discrete record of events, as a carrier with a zero and a successor function. A homomorphism between two such ledgers is a map that preserves both the zero and the successor step. The theorem hom_eq_baseRec proves that any such map from the standard counting ledger to another ledger is uniquely determined: it must send zero to the other ledger's zero, and each successor to the other ledger's successor, recursively. This is the familiar principle of definition by recursion, here proved as a formal theorem.

The theorem is a component of a larger rigidity result. Together with companion theorems, it proves that any ledger satisfying the Peano axioms (injective successor, zero not a successor, and induction) is isomorphic to the standard counting ledger. This means there is a bijective homomorphism between them, so the two structures are indistinguishable as counting systems. The framework's library proves this categorically: the standard counting ledger is the initial object in the category of such ledgers, and it is unique up to unique isomorphism.

The theorem does not claim that the natural numbers are the only possible carrier for counting in some metaphysical sense. It claims only that within the framework's definition of a ledger, any structure that satisfies the Peano axioms is isomorphic to the standard one. It also does not claim that the framework's notion of a ledger is the only way to formalize counting. Other formalizations, such as set-theoretic definitions of the natural numbers, are not addressed by this theorem. The theorem is a structural result about the framework's own definitions, not a claim about the foundations of mathematics in general.

/-- Any δ-hom out of the base agrees with `baseRec`, by δ-orbit induction using
the hom laws. This is the uniqueness half of initiality. -/
theorem hom_eq_baseRec (M : DeltaAlgebra) (g : DeltaHom deltaBase M) :
    ∀ n, g.map n = baseRec M n := by
  intro n
  induction n with
  | zero => exact g.map_zero
  | succ n ih =>
      calc g.map n.succ = M.succ (g.map n) := g.map_succ n
        _ = M.succ (baseRec M n) := by rw [ih]
        _ = baseRec M n.succ := rfl

theorem base_categorical : target_categorical :=
  fun M h => ⟨baseHom M, baseRec_injective M h, baseRec_surjective M h⟩

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 theorem does not claim that the natural numbers are the only possible carrier for counting in a metaphysical sense. The theorem does not claim that the framework's ledger is the only way to formalize counting. The theorem does not address set-theoretic or other alternative foundations of mathematics.

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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