Foundation Maximal Forcing Independence Witness

Independence witnesses

An independence witness is a pair of concrete models that settles a question of logical independence. In mathematics, a claim is independent of a set of axioms when neither it nor its negation follows from those axioms. The standard way to prove such independence is to exhibit two structures that both satisfy the axioms, one where the claim is true and one where it is false. The witness is the pair itself: two named objects, with the claim holding in one and failing in the other.

In Recognition Science, the framework's maximal forcing principle states that if a claim is not forced by the axioms, then maximal closure demands an explicit countermodel pair rather than a vague appeal to contingency. The machine-checked library of formal theorems implements this as a structure called an independence witness. The structure packages four pieces of data: a model where the claim holds, a model where it fails, and proofs that both models are admissible under the framework's admissibility rules. A theorem then converts any such witness into a formal proof that the claim is independent over the admissible class.

The plain-language content is that independence is never asserted by fiat. If the framework cannot force a claim, it must be able to point to two admissible worlds, one where the claim holds and one where it fails. This turns a negative claim about provability into a positive construction. The witness does not say the claim is false; it says the axioms are silent, and it shows exactly why by displaying both sides.

The practical effect is that the framework's library cannot smuggle in an independence claim without exhibiting the two models. A reader who wants to check an independence assertion can inspect the witness directly. This matches how working mathematicians treat independence in set theory, where forcing constructions produce explicit models, and it gives the framework the same discipline: a claim that is not forced earns its independence only by showing its two faces.

/-- 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

This module does not construct any particular independence witness for a specific claim. It does not define what admissibility means beyond the abstract membership condition. It does not prove that any given claim is independent; it only provides the conversion from witness to independence.

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)

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