Foundation Primitive Recognition Calculus Choice Principles Classical Acomega

Countable choice in the classical layer

Countable choice is a principle in mathematics that lets you make infinitely many selections at once. If you have a list of questions, one for each natural number, and each question has at least one answer, countable choice guarantees you can pick an answer for every question simultaneously. The principle is weaker than the full axiom of choice, which would let you do this for any collection of sets, not just a list indexed by natural numbers.

The declaration classical_acomega establishes that the classical display layer of the Recognition Science framework satisfies this principle. The framework's ledger, a discrete record of recognition events, uses a constructive base where such choice principles are not automatic. But the classical layer, which is deliberately chosen for certain bridge results, inherits full choice from the ambient logic. Since full choice implies countable choice, the statement is proved in one line: apply the general choice function to each question in the list.

This matters for the framework's treatment of real numbers. One of the rungs toward constructive real completeness needs a function that turns "for every precision level there exists a rational approximation" into a single infinite sequence of rational approximations. Countable choice is exactly the tool that produces such a sequence. The declaration acomega_rat_seq spells this out: given the countable choice hypothesis and a relation on rationals where every precision level has a witness, it returns a sequence of rationals satisfying every level.

What the declaration does not claim is more subtle. It does not prove countable choice from the framework's constructive base; it assumes it as a hypothesis or inherits it from the classical ambient logic. The statement itself is choice-free to write down, but it is not provable from the constructive axioms alone. The framework's audit system tracks this distinction carefully. A result that assumes countable choice as a hypothesis receives a NAMED verdict, a middle category between a conditional result and a bridge that changes the logical footing.

This calibration point matters because it keeps the framework honest about what its constructive core can do on its own. The classical layer can use full choice freely, but that power is a deliberate choice, not a consequence of the recognition calculus itself. The audit manifest records the exact footprint of every result, so readers can see precisely which results depend on choice and which do not.

/-- The classical display layer satisfies ACω: full choice specializes to
countable choice. DELIBERATELY classical (bridge tier); this theorem is the
calibration point "ACω < Classical.choice" and must not be purified. -/
theorem classical_acomega : ACOmega := fun _X _R h =>
  ⟨fun n => Classical.choose (h n), fun n => Classical.choose_spec (h n)⟩

/-- ACω yields a modulus-of-witness function for rational approximation
families: the exact shape the CRealPre completeness rung consumes (from
"every precision level has a rational witness" to a single approximation
sequence). Choice-free given the hypothesis; pure application. -/
theorem acomega_rat_seq (h : ACOmega) (R : ℕ → ℚ → Prop) (hR : ∀ n, ∃ q, R n q) :
    ∃ f : ℕ → ℚ, ∀ n, R n (f n) :=
  h ℚ R hR

/-- **ACω**, countable choice over `Type`-valued carriers. For every ℕ-indexed
family of inhabited-by-witness relations there is a simultaneous witness
function. This is the exact choice cost expected of the constructive-real
completeness rungs (Bishop/Bridges). As a `Prop` it is choice-free to STATE;
the point is that it is not provable from the δ base, while `Classical.choice`
proves it trivially (`classical_acomega`). Registered in the audit manifest's
`[registry].named` class: a clean-footprint theorem carrying `ACOmega` as a
hypothesis earns the NAMED verdict, strictly between CONDITIONAL and BRIDGE. -/
def ACOmega : Prop :=
  ∀ (X : Type) (R : ℕ → X → Prop), (∀ n, ∃ x, R n x) → ∃ f : ℕ → X, ∀ n, R n (f n)

What this page does not claim

Does not prove countable choice from the constructive base; it assumes it or inherits it from classical logic. Does not establish that the framework's constructive core can perform infinite selections without any choice principle. Does not claim that full choice is derivable from countable choice, only that full choice specializes to it.

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/PrimitiveRecognitionCalculus/ChoicePrinciples.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.

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