Foundation Maximal Forcing Primitive Selection Principle

The selection principle

A selection principle is a named reason to keep investigating a claim that is not yet settled. In the Recognition Science framework, the machine-checked library of formal theorems defines a claim as either forced, meaning it holds in every admissible realization, or independent, meaning two admissible realizations disagree on it. A claim that is neither is simply open. The SelectionPrinciple declaration gives such an open claim a label and a proposition stating why it might be worth pursuing.

The declaration does not assert that the claim is true. It does not assert that the claim is false. It does not even assert that the claim is likely. It only records that the claim is not forced and that a named selection principle exists for it. The definition of Selected makes this explicit: a claim is selected when it is not forced and there exists a selection principle for it. This is an honest tag, not a conclusion.

The purpose of the declaration is to keep the program honest. The framework's target is to derive every invariant that is invariant across all admissible realizations, and to prove every remaining degree of freedom is either forced by a deeper admissibility condition or independent by countermodel. The selection principle is the middle state: a claim that is neither forced nor independent, but has a named reason to continue working on it. It is a bookmark, not a result.

In Recognition Science, the declaration establishes a vocabulary for unfinished work. It allows the library to mark a claim as selected without pretending it is settled. This matters because the framework's larger claims rest on a chain of theorems, and an unmarked open claim could be mistaken for a proved one. The selection principle prevents that confusion by naming the claim and its reason for being pursued, while explicitly leaving its final status open.

/-- A named selection principle for claims not yet forced on the current
admissible class. -/
structure SelectionPrinciple {R : Type u} (Admissible : Set R)
    (C : RealityClaim R) where
  label : String
  applies : Prop

/-- A named selection principle for claims not yet forced on the current
admissible class. -/
structure SelectionPrinciple {R : Type u} (Admissible : Set R)
    (C : RealityClaim R) where
  label : String
  applies : Prop

/-- A claim is selected when it is not forced on the current admissible class but
does have a named selection principle. This is not final closure; it is an
honest tag that must later be strengthened to `Forced` or `Independent`. -/
def Selected {R : Type u} (Admissible : Set R) (C : RealityClaim R) : Prop :=
  ¬ Forced Admissible C ∧ Nonempty (SelectionPrinciple Admissible C)

What this page does not claim

The selection principle does not prove any claim true or false. The selection principle does not establish that any claim is forced or independent. The declaration does not assert the crown theorem of maximal forcing closure.

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

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Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: