Foundation Primitive Recognition Calculus Prccompleteness Independence Countable

The countable gap

The real numbers have a property called order-completeness: every nonempty set that is bounded above has a least upper bound, the smallest number that is still greater than or equal to everything in the set. This is what lets calculus work, since it guarantees that limits and infinite sums land back in the real numbers. A countable subfield of the reals, by contrast, always has a gap: the theorem subfield countable_subfield_not_complete proves that for any countable subfield K, there exists a nonempty set S of elements of K that is bounded above within K but has no least upper bound inside K.

The proof is short and classical. Every subfield of the reals contains the rationals, and the rationals are dense in the reals. Pick any real number r that is not in K. The set of elements of K that are less than r is nonempty and bounded above by r, but no element of K can serve as its least upper bound: a candidate below r is beaten by a rational in between, a candidate above r is not least because a rational in between already bounds the set, and r itself is not in K. Since K is countable and the reals are uncountable, such an r always exists, so the gap is unavoidable.

The framework's library proves this in a machine-checked form, and then draws a sharper conclusion. The countable field T, which is closed under the canonical cost function J(x) = (x + 1/x)/2 - 1, also lacks the least-upper-bound property. The real numbers have it. So order-completeness is not forced by the cost axioms: it holds in one cost-closed subfield, the reals, and fails in another, T. Completeness is an independent commitment, and its only model among subfields of the reals is the uncountable continuum itself.

What the theorem does not claim is that any particular construction fails, or that the framework's countable carrier is defective. It says the opposite in a precise sense: every countable carrier is incomplete, whichever one is chosen, and that is exactly what the continuum adds. The theorem also does not claim that the reals are the only complete ordered field, only that among subfields of the reals, completeness is precisely the content of uncountability.

/-- **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)

/-- ℝ has the least-upper-bound property: every nonempty bounded-above set has a least
upper bound. -/
theorem real_has_lub (S : Set ℝ) (hne : S.Nonempty) (hbdd : ∃ b, ∀ x ∈ S, x ≤ b) :
    ∃ s, IsLUB S s := by
  obtain ⟨b, hb⟩ := hbdd
  exact Real.exists_isLUB hne ⟨b, fun x hx => hb x hx⟩

/-- **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 theorem does not claim that any particular countable construction is defective or inconsistent. The theorem does not claim that the reals are the only complete ordered field in all of mathematics. The theorem does not claim that completeness is derivable from the cost axioms; it proves the opposite.

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)

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: