Foundation Primitive Recognition Calculus Prccompleteness Independence

Completeness as a separate commitment

The real numbers have a property called order-completeness: every nonempty set that has an upper bound has a least upper bound. This is the property that lets calculus work, because it guarantees that limits and suprema exist. A countable field, by contrast, can never have this property. The rational numbers are the familiar example: the set of rationals whose square is less than 2 is bounded above, but has no rational least upper bound.

In Recognition Science, the framework's ledger, a discrete record of events, is built on a countable field. The framework's library, a machine-checked collection of formal theorems, proves that any countable subfield of the reals lacks order-completeness. The theorem countable_subfield_not_complete states this directly: for any countable subfield K, there exists a nonempty set S within K that is bounded above but has no least upper bound in K. This is a theorem, not a hypothesis, because it is derived from the standard definition of a subfield and the uncountability of the reals.

The framework then applies this to its own construction. The cost function, the framework's measure of recognition effort, is closed on a specific countable field called T. The theorem completeness_not_forced_by_genuine_cost_laws shows that T satisfies the genuine cost laws, including the unit law J(1) = 0 and reciprocal symmetry J(x) = J(x⁻¹), yet T still lacks order-completeness. The real numbers, in contrast, do have it. The conclusion is that order-completeness is not a consequence of the cost axioms; it is an independent commitment that only the uncountable continuum provides.

This result matters because it draws a precise boundary. The framework's countable constructions, such as the constants and the φ-ladder, are never order-complete. If a result requires completeness, it must be added as a separate assumption, not assumed to follow from the cost laws. The result establishes this independence cleanly, separating what the framework forces from what it leaves open.

/-- **Every countable subfield of ℝ lacks the least-upper-bound property.**
Countability forces properness (ℝ is uncountable), and properness forces
incompleteness by `subfield_not_complete`. So order-completeness and countability are
flatly incompatible for subfields of ℝ: completeness is exactly what uncountability
(the continuum) buys. Whatever countable carrier δ uses, it is never order-complete. -/
theorem countable_subfield_not_complete (K : Subfield ℝ)
    (hc : (K : Set ℝ).Countable) :
    ∃ S : Set ℝ,
      (∀ x ∈ S, x ∈ K)
        ∧ S.Nonempty
        ∧ (∃ b ∈ K, ∀ x ∈ S, x ≤ b)
        ∧ ¬ ∃ s, IsLUBIn K S s := by
  refine subfield_not_complete K ?_
  intro h
  exact Cardinal.not_countable_real (h ▸ hc)

/-- **Completeness independent of the GENUINE cost laws (not just `Jcost`-closure).**
This upgrades `completeness_not_forced_by_cost_axioms`: the premise now records that
`Cost.Jcost` is a bona fide recognition cost (unit `J(1)=0`, reciprocal symmetry
`J(x)=J(x⁻¹)`, nonnegativity `J(x)≥0` on positives), and that `T` is a countable field
on which `Jcost` is closed (hence a model of those laws). `T` still fails the
least-upper-bound property while ℝ satisfies it. So order-completeness is not entailed
by the genuine cost laws plus the field structure; it is an independent commitment.
This is the credibility-gating form: a skeptic cannot say the independence rests on a
weak "closure" premise rather than the actual cost axioms. -/
theorem completeness_not_forced_by_genuine_cost_laws :
    (Cost.Jcost 1 = 0)
      ∧ (∀ x : ℝ, 0 < x → Cost.Jcost x = Cost.Jcost x⁻¹)
      ∧ (∀ x : ℝ, 0 < x → 0 ≤ Cost.Jcost x)
      ∧ (∀ x ∈ T, Cost.Jcost x ∈ T)
      ∧ (T : Set ℝ).Countable
      ∧ (∃ S : Set ℝ,
          (∀ x ∈ S, x ∈ T) ∧ S.Nonempty ∧ (∃ b ∈ T, ∀ x ∈ S, x ≤ b)
            ∧ ¬ ∃ s, IsLUBIn T S s)
      ∧ (∀ S : Set ℝ, S.Nonempty → (∃ b, ∀ x ∈ S, x ≤ b) → ∃ s, IsLUB S s) :=
  ⟨Cost.Jcost_unit0, fun _ hx => Cost.Jcost_symm hx, fun _ hx => Cost.Jcost_nonneg hx,
    fun _ hx => CostOnField.jcost_mem_T hx, T_countable, T_not_complete, real_has_lub⟩

/-- **The sharp final form.** Order-completeness is precisely the content the
continuum adds: NO countable subfield of ℝ is order-complete, while ℝ is. Since every
δ result places the carrier in a countable field (the constants, the φ-ladder, the
cost dynamics all live countably), the carrier δ uses is never order-complete,
whichever countable field it is. Completeness is an independent axiom whose only model
is uncountable. -/
theorem completeness_is_exactly_the_continuum :
    (∀ K : Subfield ℝ, (K : Set ℝ).Countable →
        ∃ S : Set ℝ, (∀ x ∈ S, x ∈ K) ∧ S.Nonempty ∧ (∃ b ∈ K, ∀ x ∈ S, x ≤ b)
          ∧ ¬ ∃ s, IsLUBIn K S s)
      ∧ (∀ S : Set ℝ, S.Nonempty → (∃ b, ∀ x ∈ S, x ≤ b) → ∃ s, IsLUB S s) :=
  ⟨countable_subfield_not_complete, real_has_lub⟩

What this page does not claim

The framework's cost function is the only possible cost function. The real numbers are the only model of the cost laws. The theorem proves that the framework's countable field is the rational numbers.

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

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