Foundation Universal Forcing Canonical Iso Equiv Of Initial Map Zero

The forced zero

In mathematics, an isomorphism between two number systems is a correspondence that preserves structure: it pairs every element of one system with an element of the other so that operations and distinguished elements line up. The simplest distinguished element is zero. A bijection, by contrast, is only a pairing of elements with no guarantee that the pairing respects anything. The gap matters: two systems can have equally many elements and still disagree about which element is zero.

The theorem equivOfInitial_map_zero closes that gap for a specific family of number systems. Within Recognition Science, the framework constructs an arithmetic from any realization of its core logical law, and between any two such arithmetics it produces a bijection. The theorem proves that this bijection sends the zero of the first forced arithmetic to the zero of the second. In plain terms: no matter which two realizations of the law you start with, the canonical correspondence between their number systems agrees on where zero lives. The proof is short: it follows directly from the defining property that one arithmetic is initial among all structures of its kind, so its map into any other is forced to preserve zero.

The companion theorem equivOfInitial_map_step extends the same preservation to the successor operation, the map that takes each number to the next one. Together the two theorems upgrade the bare bijection to a homomorphism of Peano algebras: a structure-preserving map between systems built from zero and successor. A further theorem, peanoEquiv_unique, proves that this structure-preserving isomorphism is unique: any two such isomorphisms between the same pair of forced arithmetics have the same underlying function. That uniqueness is what the word "canonical" means here, not merely that an isomorphism exists, but that there is exactly one.

In Recognition Science, this result matters because the framework's central claim is that its logical law forces structure rather than permitting it. The theorem shows that the forced arithmetic is not a loose collection of possible number systems, but a tightly constrained one: the zeros line up, the successors line up, and the correspondence between any two realizations is forced to be the same. The framework models this as a certificate: for any two realizations, there is a canonical Peano-algebra isomorphism, and it is unique.

What the theorem does not claim is just as precise. It proves preservation of zero and successor only. It does not prove that the isomorphism preserves addition, multiplication, or order; those operations are not even carried by the structure the theorem works with, which contains only zero and successor. The richer claim that the isomorphism respects the full ordered semiring structure remains open work toward the framework's Part II program. The theorem also does not assert that the forced arithmetic is the usual natural numbers; it asserts only that the correspondence between two forced arithmetics is canonical, whatever those arithmetics turn out to be.

/-- The universal forcing bijection sends the forced zero to the forced zero. -/
theorem equivOfInitial_map_zero (R : LogicRealization.{u, v}) (S : LogicRealization.{u, w}) :
    (ArithmeticOf.equivOfInitial (forcedArith R) (forcedArith S))
        (forcedArith R).peano.zero
      = (forcedArith S).peano.zero :=
  ((forcedArith R).initial.lift (forcedArith S).peano).map_zero

/-- The universal forcing bijection commutes with the forced step map. -/
theorem equivOfInitial_map_step (R : LogicRealization.{u, v}) (S : LogicRealization.{u, w})
    (x : (forcedArith R).peano.carrier) :
    (ArithmeticOf.equivOfInitial (forcedArith R) (forcedArith S))
        ((forcedArith R).peano.step x)
      = (forcedArith S).peano.step
          ((ArithmeticOf.equivOfInitial (forcedArith R) (forcedArith S)) x) :=
  ((forcedArith R).initial.lift (forcedArith S).peano).map_step x

/-- **Canonicality.**  Any two structure-preserving isomorphisms between the
forced arithmetics of two realizations have the same underlying function.  The
isomorphism furnished by Universal Forcing is therefore the unique one. -/
theorem peanoEquiv_unique (R : LogicRealization.{u, v}) (S : LogicRealization.{u, w})
    (e₁ e₂ : PeanoEquiv (forcedArith R).peano (forcedArith S).peano) :
    (e₁.toEquiv : (forcedArith R).peano.carrier → (forcedArith S).peano.carrier)
      = (e₂.toEquiv : (forcedArith R).peano.carrier → (forcedArith S).peano.carrier) :=
  (forcedArith R).initial.uniq (forcedArith S).peano e₁.toHom e₂.toHom

What this page does not claim

The theorem does not prove preservation of addition, multiplication, or order. The theorem does not assert that forced arithmetics are the usual natural numbers. The theorem does not establish that the isomorphism is unique among all bijections, only among structure-preserving ones.

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

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