Foundation Universal Forcing Strict Canonical Iso Strict Universal Forcing Peano

The forced arithmetic bridge

Peano arithmetic is the standard structure of the natural numbers: a starting point called zero, a successor operation that moves from one number to the next, and the principle that this structure is unique up to isomorphism. Any two models of this structure can be matched by a structure-preserving bijection, one that sends zero to zero and commutes with the successor map. The declaration strict_universal_forcing_peanoEquiv establishes that this same uniqueness holds inside the Recognition Science framework for a class of objects called strict realizations.

A strict realization is a minimal package of native data: comparison, composition, identity, invariance, and non-triviality rules, with no arithmetic supplied by the caller. The framework derives the arithmetic from these laws, not taking it as an input. The declaration shows that for any two strict realizations, the derived arithmetics are not merely in bijection but are isomorphic as Peano algebras: the bijection preserves zero and the successor operation. This is the structure-preserving form of the universal forcing result, upgrading a previously established bijection to a full isomorphism.

The stronger claim follows from a companion theorem, strict_peanoEquiv_unique. It asserts that any two structure-preserving isomorphisms between the forced arithmetics of two strict realizations have the same underlying function. The universal forcing isomorphism is therefore the unique one. This uniqueness is what makes the bridge canonical: no choice is involved in matching one forced arithmetic to another, because the structure itself determines the match.

In Recognition Science, this result matters because it shows that the derived arithmetic does not depend on which strict realization one starts from. Different primitive law data lead to the same arithmetic skeleton, and the identification between them is forced rather than chosen. The framework's library, a machine-checked collection of formal theorems, records this as a proved theorem with no framework-specific axioms. The practical consequence is that arithmetic, in this account, is not an arbitrary construction but a necessary output of the underlying law data.

What the declaration does not claim is equally important. It does not assert that the strict realizations themselves are unique, nor that their native data are forced. It only establishes the isomorphism between the arithmetics they generate. It also does not claim that the isomorphism extends to any additional structure beyond zero and successor, such as addition or multiplication, unless that structure is separately proved. The uniqueness result applies to structure-preserving maps, not to arbitrary functions between the carriers.

/-- **Strict Universal Forcing, structure-preserving form.**

For any two strict Law-of-Logic realizations, the bijection between their forced
arithmetics is a structure-preserving isomorphism of Peano algebras: it sends
zero to zero and commutes with the successor map. -/
noncomputable def strict_universal_forcing_peanoEquiv
    (R S : StrictLogicRealization) :
    PeanoEquiv (StrictLogicRealization.arith R).peano
      (StrictLogicRealization.arith S).peano :=
  universalForcingPeanoEquiv (StrictLogicRealization.toLightweight R)
    (StrictLogicRealization.toLightweight S)

/-- **Strict canonicality.**

Any two structure-preserving isomorphisms between the forced arithmetics of two
strict realizations have the same underlying function.  The strict universal
forcing isomorphism is therefore the unique one. -/
theorem strict_peanoEquiv_unique (R S : StrictLogicRealization)
    (e₁ e₂ : PeanoEquiv (StrictLogicRealization.arith R).peano
      (StrictLogicRealization.arith S).peano) :
    (e₁.toEquiv : (StrictLogicRealization.arith R).peano.carrier
        → (StrictLogicRealization.arith S).peano.carrier)
      = e₂.toEquiv :=
  peanoEquiv_unique (StrictLogicRealization.toLightweight R)
    (StrictLogicRealization.toLightweight S) e₁ e₂

What this page does not claim

The strict realizations themselves are not unique, only the isomorphism between their forced arithmetics. The isomorphism is not claimed to preserve addition or multiplication without separate proof. The declaration does not establish that the native law data of a strict realization is forced.

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/CanonicalIso.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.

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