Foundation Primitive Recognition Calculus Grow Delta Forced No Enumeration No En

The uncountable ledger

The declaration no_enumeration_seq establishes a classical result: there is no function from the natural numbers to the set of all functions from natural numbers to booleans that hits every such function. In plain language, you cannot make an infinite list that contains every infinite binary sequence. This is the well-known uncountability of the continuum, proved here in a constructive setting: the proof does not rely on the law of excluded middle or the axiom of choice. It works by a diagonal argument: given any proposed list, define a new sequence that differs from the nth entry at position n, so it cannot appear anywhere on the list.

The theorem is stated for binary sequences, but its force is general. It shows that the set of infinite binary sequences is not δ-enumerable, meaning no surjection from the natural numbers onto it exists. This is the choice-free heart of a broader result, ¬ DeltaForced ℝ: a carrier that is forced, in the sense of being δ-enumerable, cannot surject onto the binary sequences. The conclusion is that the continuum, the real number line, is not forced in this technical sense. The proof is short and fully checked by the machine-checked library of formal theorems; it is a theorem, not a hypothesis.

What the declaration does not claim is a statement about the real numbers directly. It does not say that the real numbers are uncountable in the usual classical sense, nor does it say anything about the cardinality of the continuum. It establishes only that no δ-enumeration of the binary sequences exists. It also does not claim that any particular construction of the real numbers fails; it is a statement about a specific notion of enumerability. The result is a stepping stone: it shows that a certain kind of forced structure cannot cover the continuum, which is a premise for further results in the framework.

/-- 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

/-- 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

The declaration does not prove that the real numbers are uncountable in the classical sense. It does not establish the cardinality of the continuum. It does not claim that any particular construction of the real numbers is impossible.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: