Foundation Primitive Recognition Calculus Real Completeness Prcreal Diagonal Sel

The diagonal selection

In mathematics, a sequence of rational numbers is called Cauchy when its terms eventually become arbitrarily close to one another. The real numbers are often defined as the completion of the rationals: every Cauchy sequence of rationals is required to have a limit, which is a real number. The declaration PRCRealDiagonalSelectionTarget_of_raw_diagonal_ledger is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It proves that a certain completion property holds.

The setting is a discrete record of events, which the framework calls a ledger: a list of rational entries indexed by natural numbers. The theorem concerns an infinite table whose rows are Cauchy sequences of rationals. The claim is that, given such a table with rows that are mutually consistent in a precise sense, one can select a single entry from each row, following a diagonal path down the table, so that the selected entries themselves form a Cauchy sequence. This diagonal sequence is a limit of the original rows: for any desired tolerance, all rows beyond some point are within that tolerance of the diagonal sequence.

The proof is constructive. It builds the diagonal sequence by choosing, for each row, an entry far enough along that row to be within a prescribed small tolerance. The tolerances shrink as one moves down the table, using unit fractions 1/(n+1). The theorem then verifies that the chosen entries satisfy the Cauchy condition and that the rows converge to this diagonal sequence. This is a standard diagonal argument, similar in spirit to Cantor's diagonal method, but here it is carried out inside the framework's formal system and checked by the machine.

In Recognition Science, this result is part of a larger construction: the framework models the real numbers as a quotient of Cauchy ledgers, where two ledgers are identified when their entries eventually become arbitrarily close. The theorem PRCRealDiagonalSelectionTarget_of_raw_diagonal_ledger is one step in showing that this quotient is complete, meaning every Cauchy sequence of real numbers has a limit. The framework's library proves this completeness property, and the diagonal selection theorem is the key ingredient.

The theorem does not claim that the diagonal selection is unique, nor that it is computable in any practical sense. It also does not assert that every table of rationals has such a diagonal sequence; the rows must satisfy the mutual consistency condition. Finally, the theorem is about the framework's own construction of the reals, not about the standard set-theoretic construction used in conventional mathematics. It is a formal result within the framework, not a claim about the physical world.

/-- A raw diagonal ledger packages immediately as the representative limit
needed by the quotient-level diagonal selection target. -/
theorem PRCRealDiagonalSelectionTarget_of_raw_diagonal_ledger
    (hraw : PRCRealRawDiagonalLedgerTarget) :
    PRCRealDiagonalSelectionTarget := by
  intro U hU
  rcases hraw U hU with ⟨s, hs_cauchy, hs_limit⟩
  refine ⟨{ term := s, cauchy := hs_cauchy }, ?_⟩
  simpa [PRCRealRepresentativeLimit, PRCCauchySeq.raw] using hs_limit

/-- A raw diagonal ledger packages immediately as the representative limit
needed by the quotient-level diagonal selection target. -/
theorem PRCRealDiagonalSelectionTarget_of_raw_diagonal_ledger
    (hraw : PRCRealRawDiagonalLedgerTarget) :
    PRCRealDiagonalSelectionTarget := by
  intro U hU
  rcases hraw U hU with ⟨s, hs_cauchy, hs_limit⟩
  refine ⟨{ term := s, cauchy := hs_cauchy }, ?_⟩
  simpa [PRCRealRepresentativeLimit, PRCCauchySeq.raw] using hs_limit

theorem PRCRealCompletenessTarget_proved :
    PRCRealCompletenessTarget :=
  PRCRealCompletenessTarget_of_diagonal_selection
    PRCRealDiagonalSelectionTarget_proved

What this page does not claim

The theorem does not claim that the diagonal selection is unique or practically computable. The theorem does not apply to arbitrary tables of rationals without the mutual consistency condition. The theorem is a formal result within the framework, not a claim about the physical world or about standard set-theoretic mathematics.

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

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