Foundation Primitive Recognition Calculus Generable Real Rat Mem

The rationals are always reachable

A rational number is a ratio of two whole numbers, such as 2/3 or 355/113. The declaration rat_mem states that every rational number belongs to the generable reals: the collection of all real numbers obtainable from a countable family of named constants and the rationals by finitely many additions, multiplications, subtractions, and divisions. This is a theorem in the machine-checked library of formal theorems, proved for any countable constant family κ and any rational q.

The proof is immediate from the definition: the generable reals form a subfield of the real numbers, and every subfield must contain the prime field, which is exactly the rationals. The declaration rat_mem records this fact formally. It is a small piece of a larger framework in which the generable reals serve as the operational carrier for what can be finitely described. The framework models reality as storing a discrete record of recognition events, and the generable reals are the numbers that such a record can actually name.

The theorem's force is negative as well as positive. Because the generable reals are countable, they form a proper subset of the real numbers: there are real numbers that no finite description can reach. The framework proves that its analysis display reaches the whole continuum, so the gap between what can be displayed and what can be generated is exactly the reals that exist only as display, never as finite generation. This guards against smuggling uncountable ontology in through the analysis interface.

What rat_mem does not claim is equally precise. It does not say that every rational is a named constant; it says every rational is generable from the constants by field operations. It does not say that the generable reals are all the reals; the same file proves they are a proper subset. And it does not say that the rationals are the only generable numbers; the named constants themselves are generable, and so are all finite field combinations of them.

/-- Every rational is generable (the prime field sits inside). -/
theorem rat_mem (κ : ℕ → ℝ) (q : ℚ) : (q : ℝ) ∈ genField κ :=
  SubfieldClass.ratCast_mem (genField κ) q

/-- The generable reals are a proper subset of ℝ: countability rules out the whole
continuum. -/
theorem genField_proper (κ : ℕ → ℝ) : (genField κ : Set ℝ) ≠ Set.univ := by
  intro h
  exact Cardinal.not_countable_real (h ▸ genField_countable κ)

/-- **Phase 2 headline: display exceeds generation.** For any countable constant
family, there is a real that is the value of a Delta-real protocol (so the
analysis display reaches it) yet is not generable. The protocol value map lands
onto the full continuum, while the ontology is the countable generable field; the
gap is exactly the reals that exist only as display, never as finite generation.
This is the guard against smuggling uncountable ontology in through the analysis
interface. -/
theorem display_exceeds_generation (κ : ℕ → ℝ) :
    ∃ r : ℝ, (∃ x : DeltaReal.Protocol, x.value = r) ∧ r ∉ genField κ := by
  obtain ⟨r, hr⟩ := (Set.ne_univ_iff_exists_notMem _).mp (genField_proper κ)
  exact ⟨r, DeltaReal.Protocol.value_surjective r, hr⟩

What this page does not claim

The theorem does not claim that every rational is a named constant. It does not claim that the generable reals are all the real numbers. It does not claim that the rationals are the only generable 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/GenerableReal.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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