Foundation Reals From Logic Eq Iff To Real Eq

The equality bridge

The real numbers are the complete ordered field that fills the gaps between rationals. In the Recognition Science framework, a construction called LogicReal rebuilds the reals from a recovered rational layer, wrapping Mathlib's Bourbaki completion of the rationals. The declaration eq_iff_toReal_eq is the bridge that connects this recovered real type to the standard real numbers: it states that two LogicReal values are equal if and only if their images under the map toReal are equal. This is the fundamental identification that lets every theorem about ordinary reals be pulled back into the framework.

The declaration is a transport principle. Algebra and order on LogicReal are defined by pulling back the corresponding structures along toReal, so every downstream statement can reduce to the existing real statement and then be read back as a recovered-real statement. The declaration eq_iff_toReal_eq is the keystone of this transport-first API: it says the map toReal is injective, so no information is lost when moving between the two worlds. The proof is a simple consequence of the definitions, but its role is structural, not computational.

In Recognition Science, the ledger (a discrete record of events) gives rise to rationals, and the completion of those rationals gives rise to the reals. The declaration eq_iff_toReal_eq is the formal statement that this recovered real type is not a new kind of number, but a faithful copy of the standard reals. It does not claim that the framework creates a new real number system, nor that it proves the existence of the reals from scratch. It merely establishes that the recovered reals behave exactly like the ordinary reals, so that all the usual real analysis applies.

What the declaration does not claim is equally important. It does not say that the recovered reals are the only possible completion of the rationals, nor that the construction is simpler than the standard one. It does not assert that every real number is reachable from the ledger, only that the recovered reals are isomorphic to the standard reals. The declaration is a transport result, not a foundational claim about the nature of the continuum.

/-- `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

The declaration does not prove that the recovered reals are the only possible completion of the rationals. It does not assert that every real number is reachable from the ledger. The declaration is a transport result, not a foundational claim about the nature of the continuum.

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)

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

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