Foundation Reals From Logic

The recovered real numbers

The real numbers are the complete ordered field that fills the gaps between rational numbers. A real number like √2 or π cannot be written as a fraction of two integers, but it can be approached arbitrarily closely by fractions: 1.4, 1.41, 1.414, and so on. The standard way to define reals is as limits of such rational sequences, and this is what the construction uses.

The history of this idea runs from ancient Greek approximations through Richard Dedekind's cuts and Georg Cantor's Cauchy sequences in the 1870s. Cantor's construction, named after Augustin-Louis Cauchy's convergence criterion, defines a real number as an equivalence class of rational sequences that get closer together. This construction follows that path, but the rationals it starts from are themselves recovered from logic.

In Recognition Science, the framework first derives the natural numbers from the Law of Logic, then integers, then rationals, and finally reals. The type LogicReal, a discrete record of events that the framework uses to track recognition, is a wrapper around the completion of the rationals. The key map toReal is the canonical comparison equivalence with the standard Cauchy real numbers, so every theorem about ordinary reals transfers directly.

The construction is transport-first: algebra and order on LogicReal are defined by pulling back the corresponding structures along toReal. This means addition, multiplication, and ordering on the recovered reals behave exactly as they do on the standard reals. The construction proves the field axioms, including commutativity, associativity, distributivity, and the existence of inverses, all by reducing to the existing real theorems.

What this establishes in plain language is that the real numbers are not an extra assumption. They follow from the same logical foundation that produces the natural numbers, so the continuum of analysis and physics is recovered rather than postulated. The framework's library, a machine-checked collection of formal theorems, verifies this chain from logic to reals without any framework-specific axioms.

/-- `LogicReal` is the Cauchy completion of the recovered rationals, realized
through the canonical completion of `ℚ` and the equivalence `LogicRat ≃ ℚ`.

The wrapper prevents global instance pollution on `Completion ℚ` while still
letting us reuse Mathlib's completed real line. -/
structure LogicReal where
  val : CompareReals.Bourbakiℝ

/-- Transport a recovered real to Mathlib's real line. -/
noncomputable def toReal (x : LogicReal) : ℝ :=
  CompareReals.compareEquiv x.val

/-- `LogicReal` is the Cauchy completion of the recovered rationals, realized
through the canonical completion of `ℚ` and the equivalence `LogicRat ≃ ℚ`.

The wrapper prevents global instance pollution on `Completion ℚ` while still
letting us reuse Mathlib's completed real line. -/
structure LogicReal where
  val : CompareReals.Bourbakiℝ

/-- `LogicReal` is the Cauchy completion of the recovered rationals, realized
through the canonical completion of `ℚ` and the equivalence `LogicRat ≃ ℚ`.

The wrapper prevents global instance pollution on `Completion ℚ` while still
letting us reuse Mathlib's completed real line. -/
structure LogicReal where
  val : CompareReals.Bourbakiℝ

What this page does not claim

This construction does not derive the real numbers from scratch; it uses Mathlib's completion of the rationals as the engine. The framework does not claim that the real numbers are the only possible continuum, only that this construction recovers them.

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/RealsFromLogic.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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