Foundation Domain Bootstrap Comparison Operator On

The comparison operator

A comparison operator is a function that takes two numbers and returns a third number. In the Recognition Science framework, it is written as a function of two arguments, and it is the primitive from which the framework's cost function is derived: the cost of recognizing a value is the comparison operator applied to that value and 1. The declaration ComparisonOperatorOn is an abbreviation, a definitional shorthand, for the type of such functions on a generic linearly ordered field, meaning a set equipped with addition, multiplication, and a total order that respects them.

The classical fact that anchors this declaration is the standard characterization of the real numbers as the unique Archimedean Dedekind-complete ordered field. This is a classical theorem: among all ordered fields, the real numbers are the only one that is both Archimedean, meaning no infinite elements and no infinitesimals, and Dedekind-complete, meaning every nonempty set with an upper bound has a least upper bound. The framework's library, a machine-checked collection of formal theorems, uses this classical fact to prove a bootstrap theorem.

In Recognition Science, the framework models the Law of Logic as a comparison operator satisfying four Aristotelian conditions: identity, non-contradiction, scale invariance, and distinguishability. The declaration LogicSupported packages these conditions into a structure on a generic ordered field. The bootstrap theorem then states that if such a field is Archimedean and conditionally complete, it is canonically isomorphic to the real numbers as an ordered field. This closes a chicken-and-egg: the framework uses the real numbers as its ambient field, and the theorem proves that any other field that could support the same structure must be the same field, up to isomorphism.

The theorem does not claim that the Law of Logic alone forces the real numbers. The Archimedean and completeness hypotheses are named explicitly as residual analytic inputs; without them, the comparison operator could live on the rationals or any incomplete subfield. The framework's own summary states that the single residual classical input is the Archimedean completeness of the ambient field, and that this input is named, not hidden. What the theorem does establish is that the choice of the real numbers as the domain for comparison operators is canonical up to isomorphism, not contingent.

/-- A comparison operator on a linearly ordered field. -/
abbrev ComparisonOperatorOn (K : Type*) := K → K → K

/-- **Bootstrap theorem (named-hypothesis form)**: a linearly ordered
field on which the Law of Logic is supported and which is Archimedean
and conditionally complete is canonically isomorphic to `ℝ` as an
ordered field. The Archimedean and conditional-completeness
hypotheses are the analytic content the Law of Logic does not on its
own provide; they are named here as inputs.

The conclusion is the existence of an order-preserving ring
isomorphism with `ℝ`. -/
theorem bootstrap_to_real
    (K : Type*) [ConditionallyCompleteLinearOrderedField K]
    (_ : LogicSupported K) :
    Nonempty (K ≃+*o ℝ) :=
  ⟨LinearOrderedField.inducedOrderRingIso K ℝ⟩

/-- **Bootstrap theorem (named-hypothesis form)**: a linearly ordered
field on which the Law of Logic is supported and which is Archimedean
and conditionally complete is canonically isomorphic to `ℝ` as an
ordered field. The Archimedean and conditional-completeness
hypotheses are the analytic content the Law of Logic does not on its
own provide; they are named here as inputs.

The conclusion is the existence of an order-preserving ring
isomorphism with `ℝ`. -/
theorem bootstrap_to_real
    (K : Type*) [ConditionallyCompleteLinearOrderedField K]
    (_ : LogicSupported K) :
    Nonempty (K ≃+*o ℝ) :=
  ⟨LinearOrderedField.inducedOrderRingIso K ℝ⟩

What this page does not claim

The Law of Logic on its own forces the ambient field to be the real numbers. The comparison operator is defined on a specific field rather than on a generic linearly ordered field. The framework derives the Archimedean completeness of the real numbers from its own axioms.

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/DomainBootstrap.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.

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