Foundation Primitive Recognition Calculus Delta Forced Delta Forced Prod

The product of forced types

A countable set is one whose members can be matched, one by one, against the natural numbers 0, 1, 2, and so on. The rational numbers are countable, the integers are countable, and so is any finite collection. The real numbers are not: Georg Cantor proved in 1874 that no such listing can cover them all. Countability is the mathematical core of delta-forced, the framework's term for a type that carries an explicit injection into the naturals, meaning a concrete rule that assigns a distinct natural number to each element.

The declaration deltaForced_prod proves a closure property: if two types X and Y are each delta-forced, then their product type X × Y, the set of all ordered pairs (x, y), is also delta-forced. The proof is short and constructive. It first notes that a delta-forced type is countable, then invokes the standard fact that the Cartesian product of two countable sets is countable. The theorem is tagged THEOREM in the machine-checked library, meaning its statement is fully verified by the proof checker with no unproved assumptions.

The framework gives this plain mathematical fact a physical reading. Its thesis identifies physically real with delta-forced: a system is real, in this account, exactly when its states can be enumerated. Under that identification, the product theorem says that if two systems are each physically real, then the combined system of their paired states is physically real as well. The theorem deltaForced_sum states the analogous closure for disjoint unions, and deltaForced_subtype for subsets. Together they form a small algebra of what the framework calls the forced, or enumerable, universe.

The theorem does not claim that every countable set is physically real, nor that the identification itself is proved. The identification is a definitional choice, recorded in the declaration PhysicallyReal, which simply abbreviates delta-forced. The mathematical content is the countability closure; the physical gloss is a separate thesis. The framework's library also proves the demarcation theorem: the naturals, integers, and rationals are delta-forced, while the reals are not, a result that follows from Cantor's diagonal argument.

/-- Pairing two forced collections is forced. -/
theorem deltaForced_prod {X : Type u} {Y : Type v}
    (hX : DeltaForced X) (hY : DeltaForced Y) : DeltaForced (X × Y) := by
  have : Countable X := countable_of_deltaForced hX
  have : Countable Y := countable_of_deltaForced hY
  exact (deltaForced_iff_countable _).mpr inferInstance

/-- A δ-forced type is countable (the certificate is an injection into ℕ).
Choice-free. -/
theorem countable_of_deltaForced {X : Type u} (h : DeltaForced X) : Countable X := by
  obtain ⟨e⟩ := h
  exact e.injective.countable

/-- **Demarcation.** The δ tower (ℕ, ℤ, ℚ) is physically real; the continuum ℝ is
not. The forced-tower conjuncts are choice-free (`forcedTower`); the ℝ conjunct uses
the classical uncountability of ℝ, which is a fact about the display-tier object, not
about the forced side. -/
theorem demarcation :
    PhysicallyReal ℕ ∧ PhysicallyReal ℤ ∧ PhysicallyReal ℚ ∧ ¬ PhysicallyReal ℝ :=
  ⟨deltaForced_nat, deltaForced_int, deltaForced_rat, not_deltaForced_real⟩

/-- The ontological reading: **physically real** is, by thesis, exactly δ-forced. The
mathematical content is carried entirely by `DeltaForced`; this name records the
claim that the demarcation line below is the physical one. -/
def PhysicallyReal (X : Type u) : Prop := DeltaForced X

What this page does not claim

The theorem does not prove that every countable set is physically real. The physical identification of real with enumerable is a definitional choice, not a proved result. The theorem does not address whether the product of two uncountable types could be delta-forced.

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

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