Foundation Maximal Forcing Independence Witness Independent Of Witness

Witnessing independence

In mathematics, showing that a statement is independent of a theory means showing that it can be neither proved nor disproved from the theory's axioms. The standard method is to build two models of the theory: one in which the statement is true, and one in which it is false. The Recognition Science framework, a machine-checked library of formal theorems, applies this same standard to its own claims about reality. Its recognition ledger, a discrete record of events, must not leave a claim's status vague when the claim is not forced.

The declaration independent_of_witness is a theorem that converts an explicit pair of models into a formal independence tag. The pair consists of two realizations of the claim universe, one admissible model where the claim holds and one admissible model where it fails. Admissibility here means each model belongs to the class of models the framework accepts as legitimate. The theorem then assembles these four facts, two admissibility proofs and two satisfaction proofs, into the proposition-level Independent tag. In plain terms, if you can exhibit both a yes-model and a no-model, the framework formally records the claim as independent.

The theorem does not itself produce the two models. It only packages them once they exist. Nor does it say anything about which claims are independent; it is a bridge from a concrete witness to a formal tag. The work of finding the models, and of proving they are admissible, lies entirely with the person or process that supplies the witness. The declaration's role is to make the independence status explicit and machine-checked, rather than a matter of informal judgment.

What the declaration does not claim is equally precise. It does not assert that any particular claim is independent. It does not claim that a claim without a witness is dependent; absence of a witness is not a proof either way. It does not define what counts as admissible; that is a separate structure in the framework. The theorem is a conditional bridge, not a discovery procedure.

/-- An explicit witness implies the proposition-level `Independent` tag. -/
theorem independent_of_witness {U : ClaimUniverse.{u}}
    {C : RealityClaim U.Realization}
    (W : IndependenceWitness U C) :
    Independent U.admissibility.admissible C := by
  exact ⟨W.yes_model, W.no_model, W.yes_admissible, W.no_admissible,
    W.yes_holds, W.no_fails⟩

/-- Explicit countermodel pair for independence of a claim over an admissible
class. -/
structure IndependenceWitness (U : ClaimUniverse.{u})
    (C : RealityClaim U.Realization) where
  yes_model : U.Realization
  no_model : U.Realization
  yes_admissible : yes_model ∈ U.admissibility.admissible
  no_admissible : no_model ∈ U.admissibility.admissible
  yes_holds : C.holds yes_model
  no_fails : ¬ C.holds no_model

What this page does not claim

No particular claim is asserted to be independent. A claim without a witness is not thereby dependent. The declaration does not define admissibility.

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/MaximalForcing/IndependenceWitness.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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