Foundation Primitive Recognition Calculus Grow Delta Forced No Enumeration

The enumeration barrier

The recognition ledger, a discrete record of events, grows by forced steps. A natural question is whether such a process could eventually enumerate every possible infinite binary sequence, the kind of object that makes up the continuum. The framework's library, a machine-checked collection of formal theorems, answers this with a definitive no.

The core result is the constructive Cantor theorem: the space of binary sequences ℕ → Bool cannot be δ-enumerated. In plain language, no function from the natural numbers can list all infinite sequences of true and false values. This is proved by a diagonal argument: any proposed listing f fails at the sequence that flips the k-th bit of the k-th entry. The theorem is formalized in the declaration no_enumeration_seq, with no choice axioms needed.

This barrier has a direct consequence for the framework's own growth. A forced (δ-enumerable) carrier cannot surject onto the binary sequences. Since the continuum would require such a listing, the continuum is not forced. The framework's recognition process, however it grows, cannot reach the full set of infinite binary sequences.

What this establishes is a boundary condition on what Recognition Science can claim to derive. The framework proves that its own growth mechanism is strictly weaker than the full continuum. This is not a failure but a precise structural fact: the ledger's reach is limited by the same diagonal argument that underlies Cantor's theorem.

/-- The constructive Cantor theorem: the binary-sequence space `ℕ → Bool` cannot be
δ-enumerated. This is the choice-free heart of `¬ DeltaForced ℝ`: a forced
(δ-enumerable) carrier cannot surject onto the binary sequences, so the continuum
is not forced. -/
theorem no_enumeration_seq : ¬ ∃ f : ℕ → (ℕ → Bool), Function.Surjective f := by
  rintro ⟨f, hf⟩
  obtain ⟨n, hn⟩ := hf (fun k => !(f k k))
  have := congrFun hn n
  simp at this

/-- The constructive Cantor theorem: the binary-sequence space `ℕ → Bool` cannot be
δ-enumerated. This is the choice-free heart of `¬ DeltaForced ℝ`: a forced
(δ-enumerable) carrier cannot surject onto the binary sequences, so the continuum
is not forced. -/
theorem no_enumeration_seq : ¬ ∃ f : ℕ → (ℕ → Bool), Function.Surjective f := by
  rintro ⟨f, hf⟩
  obtain ⟨n, hn⟩ := hf (fun k => !(f k k))
  have := congrFun hn n
  simp at this

What this page does not claim

This module does not prove that the continuum exists or that it is uncountable in any classical sense. It does not claim that the recognition process is finite or bounded in length. It does not establish any upper bound on what the framework can derive about individual real 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/Grow/DeltaForcedNoEnumeration.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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