Foundation Maximal Forcing Primitive Independent

What independence means

In mathematics and physics, a statement is independent of a set of axioms when neither it nor its negation follows from those axioms. The Recognition Science framework formalizes this idea for claims about reality. Its ledger, a discrete record of events, is the object under study. The framework's recognition events, the recorded distinctions that reality keeps track of, are what claims describe.

The declaration Independent in the framework's machine-checked library of formal theorems defines this precisely. A claim is independent over an admissible class when there exist two admissible realizations, two ways the ledger could be, such that the claim holds in one and fails in the other. This is a constructive definition: it does not merely say a proof is missing, it exhibits two concrete alternatives that disagree. The definition appears in the module MaximalForcing/Primitive.lean as the declaration Independent.

The definition works alongside two companions. Forced means a claim holds in every admissible realization, so no alternative can contradict it. Selected marks a claim that is not yet forced but has a named selection principle, an honest intermediate tag that must later be strengthened to either forced or independent. The Independent declaration is the third status, the one that says the framework has found a genuine fork in the road.

In Recognition Science, the program's target is to derive every invariant that holds across all admissible realizations, and to prove every remaining degree of freedom is either forced by a deeper condition or independent by countermodel. The Independent declaration is the tool that closes the second half of that target. It does not assert that any particular claim is independent, only what independence would mean. The module that contains it intentionally does not assert the crown theorem.

What the declaration does not claim is as important as what it does. It does not claim that any specific claim about reality is independent, only that the concept is well-defined. It does not claim that an independent claim is false, only that it is not forced. And it does not claim that the two admissible realizations are physically possible, only that they are admissible under the framework's own rules.

/-- A claim is independent over an admissible class when two admissible
realizations disagree on it. -/
def Independent {R : Type u} (Admissible : Set R) (C : RealityClaim R) : Prop :=
  ∃ R0 R1 : R,
    R0 ∈ Admissible ∧ R1 ∈ Admissible ∧ C.holds R0 ∧ ¬ C.holds R1

/-- A claim is forced on an admissible class when it holds in every admissible
realization. -/
def Forced {R : Type u} (Admissible : Set R) (C : RealityClaim R) : Prop :=
  ∀ R0 : R, R0 ∈ Admissible -> C.holds R0

/-- 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

No specific claim about reality is asserted to be independent. Independent claims are not asserted to be false, only not forced. The two admissible realizations are not asserted to be physically possible.

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)

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

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: