Foundation Primitive Recognition Calculus Frscarrier Carrier Values Countable

The countable carrier

The real numbers are uncountable: there are so many of them that no list, however infinite, can contain them all. The framework's carrier, the set of values its basic arithmetic can actually produce, is different. It is countable, meaning its members can be arranged in a single infinite list. This is not a philosophical preference but a proved fact in the machine-checked library of formal theorems.

The carrier is built from a finite syntax. It starts with the rational numbers and four named constants: the golden ratio φ, π, e, and the inverse fine-structure constant α⁻¹. From these, the syntax allows negation, addition, multiplication, and division (by nonzero values). Every expression is a finite string of symbols, and only countably many finite strings exist. The theorem carrierValues_countable proves that the set of all values these expressions evaluate to is countable. A companion theorem shows this set is a proper subset of the reals, so it is not everything.

The proof works by showing every carrier value lies inside a known countable field, the rsField. Since the field is countable and the carrier is a subset of it, the carrier inherits countability. The same library also proves that every carrier value has a representation as a protocol in the Δ-real interface, meaning the finite syntax is enough to display each value computationally.

What this does not claim is important. Countability does not mean the carrier is small in any practical sense; it still contains infinitely many values. It does not say which real numbers are missing, only that some are. It does not assert that the framework's physics is limited to these values, nor that the constants named are the only ones allowed. The theorem is about the syntax's expressive power, not about the physical world.

In Recognition Science, this result matters because it pins down what the framework actually computes on. The carrier is finite generation over a fixed inventory, not the uncountable continuum. That distinction is a structural fact about the formal system, proved once and for all in the library.

/-- The carrier value set is countable: only countably many finite terms exist. -/
theorem carrierValues_countable : carrierValues.Countable :=
  (MinimalField.rsField_countable).mono carrierValues_subset

/-- The carrier values are a proper subset of ℝ. -/
theorem carrierValues_proper : carrierValues ≠ Set.univ := by
  intro h
  exact Cardinal.not_countable_real (h ▸ carrierValues_countable)

/-- **Soundness.** Every carrier term evaluates into the countable field `rsField`.
The finite-description syntax never escapes the carrier. -/
theorem eval_mem (ex : FRSExpr) : eval ex ∈ MinimalField.rsField := by
  induction ex with
  | rat q => exact SubfieldClass.ratCast_mem MinimalField.rsField q
  | phi => simpa [eval] using MinimalField.rsField_mem_phi
  | pi => simpa [eval] using MinimalField.rsField_mem_pi
  | e => simpa [eval] using MinimalField.rsField_mem_e
  | alphaInv => simpa [eval] using MinimalField.rsField_mem_alphaInv
  | neg a ih => simpa [eval] using neg_mem ih
  | add a b iha ihb => simpa [eval] using add_mem iha ihb
  | mul a b iha ihb => simpa [eval] using mul_mem iha ihb
  | inv a ih => simpa [eval] using inv_mem ih

What this page does not claim

The carrier contains only finitely many values. The framework's physical models are limited to carrier values. The theorem identifies which real numbers are missing from the carrier.

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

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