Foundation Universal Forcing Strict Mathlib Nno Mathlib Nnocert Holds

The natural-number certificate

Natural numbers are the counting numbers 0, 1, 2, 3, and so on, with no gaps and no repetitions. They have a defining property called recursion: if you give a starting value and a rule for stepping from one value to the next, you get exactly one sequence. For example, starting at 1 and stepping by doubling gives 1, 2, 4, 8, and nothing else. This property, that a starting point plus a step rule determines a unique sequence, is what makes the natural numbers a natural number object, or NNO, in category theory.

Recognition Science builds its own version of the counting numbers, called LogicNat, from its foundational ledger of recognition events. The question is whether these framework-internal numbers behave like the ordinary natural numbers of standard mathematics. The declaration mathlibNNOCert_holds answers that question. It is a theorem in the framework's machine-checked library of formal theorems, and it states that LogicNat satisfies the recursion property: for any type α, any base element, and any step function, there exists exactly one function from LogicNat to α that sends zero to the base and succ to the step. The certificate bundles two facts: existence, meaning such a function always exists, and uniqueness, meaning there is only one such function.

In Recognition Science, this certificate is the bridge that connects the framework's internal counting numbers to the natural numbers used throughout standard mathematics. It means that when a theorem in the framework talks about counting, it is talking about the same kind of counting that appears in ordinary arithmetic and number theory. The framework's numbers are not a strange parallel universe; they satisfy the same fundamental recursion property that defines the natural numbers in category theory.

What the certificate does not claim is equally important. It does not claim that LogicNat is the only possible natural number object, nor does it claim anything about the physical world. It is a statement about the internal structure of the framework, showing that its counting numbers are well-behaved. It does not say that the framework's ledger is the real universe's ledger, and it does not prove any empirical fact about measurements. The certificate is a piece of pure mathematics, establishing a structural correspondence between two formal systems.

theorem mathlibNNOCert_holds : MathlibNNOCert :=
{ exists_rec := logicNat_has_type_NNO_universal_property
  unique_rec := logicNat_NNO_uniqueness }

theorem logicNat_has_type_NNO_universal_property :
    ∀ {α : Type*} (base : α) (step : α → α),
      ∃ (f : LogicNat → α),
        f LogicNat.zero = base ∧
        ∀ n, f (LogicNat.succ n) = step (f n) :=
  @nno_universal_existence

theorem logicNat_NNO_uniqueness :
    ∀ {α : Type*} (base : α) (step : α → α)
      (f g : LogicNat → α),
      f LogicNat.zero = base → (∀ n, f (LogicNat.succ n) = step (f n)) →
      g LogicNat.zero = base → (∀ n, g (LogicNat.succ n) = step (g n)) →
      f = g :=
  @nno_universal_uniqueness

What this page does not claim

LogicNat is the only possible natural number object in category theory. The certificate proves any empirical fact about physical measurements. The framework's ledger is identical to the real universe's ledger.

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

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