Foundation Universal Forcing Strict Categorical Strict Categorical Arith Equiv L

The arithmetic carrier

The natural numbers are the counting numbers 0, 1, 2, 3, and so on. In the Recognition Science framework, the declaration strictCategorical_arith_equiv_logicNat establishes that the natural-number object built from categorical logic is the same as the framework's own ledger, a discrete record of events. The identification is a proved equivalence, not a choice: the two structures are interchangeable for all arithmetic purposes.

The declaration is a definition in the machine-checked library of formal theorems. It states that the carrier of the Peano arithmetic realization, obtained from a strict categorical construction, is equivalent to LogicNat, the framework's canonical natural-number surface. The equivalence is built from an orbit equivalence, meaning that every counting position in one structure corresponds to exactly one in the other, and the correspondence respects the successor operation.

The practical content is that the framework's arithmetic does not float free. It is anchored to a standard categorical presentation of the natural numbers, the kind a logician would recognize. This matters because the framework's later results, such as the forcing chain that derives constants and dimensions, rely on arithmetic as a substrate. The declaration supplies that substrate with a precise, machine-checked link.

In Recognition Science, the framework models cost as a function on pairs of ledger entries. The declaration does not itself prove any property of that cost function. It only identifies the counting structure on which cost is defined. The cost function's symmetry and self-cost-zero properties are separate theorems, proved elsewhere in the library.

The equivalence also does not claim that the categorical construction is the only way to present the natural numbers, nor that it is the most convenient. It claims only that this particular presentation and the framework's native one are the same object. A reader who wants to know why the framework chose this presentation will find that question open; the declaration itself is silent on motivation.

def strictCategorical_arith_equiv_logicNat :
    (StrictLogicRealization.arith strictCategoricalRealization).peano.carrier
      ≃ LogicNat :=
  (StrictLogicRealization.toLightweight strictCategoricalRealization).orbitEquivLogicNat

def strictCategorical_arith_equiv_logicNat :
    (StrictLogicRealization.arith strictCategoricalRealization).peano.carrier
      ≃ LogicNat :=
  (StrictLogicRealization.toLightweight strictCategoricalRealization).orbitEquivLogicNat

def strictCategorical_arith_equiv_logicNat :
    (StrictLogicRealization.arith strictCategoricalRealization).peano.carrier
      ≃ LogicNat :=
  (StrictLogicRealization.toLightweight strictCategoricalRealization).orbitEquivLogicNat

What this page does not claim

The declaration does not prove the cost function's symmetry or self-cost-zero properties. The declaration does not claim that this categorical presentation is the only or most convenient way to present the natural numbers. The declaration does not derive any physical constant or dimension.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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