Foundation Primitive Recognition Calculus Frscarrier Rat Is Term

The rationals as terms

The declaration rat_is_term is a small but load-bearing result in the Recognition Science framework's formal library, a machine-checked collection of theorems. It states that for any rational number q, the expression FRSExpr.rat q evaluates to exactly the real number q. In plainer terms: if the framework's language of expressions includes a slot for a rational number, then putting a rational number in that slot and evaluating the expression gives you back precisely that rational number, no more and no less.

This theorem is part of a larger definition: the carrier, the set of values the framework's computations can actually produce. The carrier is built from a finite inventory of constants: the golden ratio φ, π, e, and the inverse fine-structure constant α⁻¹, plus the rationals. These are combined using only addition, multiplication, and inversion. The theorem rat_is_term is the first step in showing that this syntax is sound: every expression built this way evaluates to a real number that belongs to a specific countable field, a set of numbers closed under the basic arithmetic operations.

What makes this theorem notable is not the equation itself, which is almost a definition, but what it enables. Because every rational is a term, and because the other constants are also terms, the entire carrier is countable: there are only countably many finite expressions, so the carrier can only contain countably many values. A separate theorem proves this set is a proper subset of the real numbers, meaning the framework's computations never reach the uncountable continuum. The carrier is a finite-generation structure over a fixed inventory, not the full real line.

In Recognition Science, this matters because the framework models reality as preserving a discrete record of recognition events. The carrier is the vocabulary that record can use. The theorem rat_is_term guarantees that the vocabulary includes every rational number, so the ledger can record any rational value exactly. This is a foundational assurance: the framework's language is not missing a basic class of numbers.

The theorem does not claim that every real number is a term, and it does not claim that the carrier includes irrational numbers beyond the four named constants. It also does not claim that the rationals are the only terms. It is a precise, narrow statement about one constructor in an inductive type, proved by reflexivity in the formal library.

theorem rat_is_term (q : ℚ) : eval (FRSExpr.rat q) = (q : ℝ) := rfl

/-- 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)

What this page does not claim

Not every real number is a term in the carrier. The theorem does not claim that the rationals are the only terms. The carrier being countable does not mean it is finite.

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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