Foundation Domain Bootstrap Required

The bootstrap theorem

The real numbers, denoted ℝ, are the unique complete ordered field. This classical fact, known since the 19th century, states that any number system with the same algebraic structure, order, and a completeness property (no gaps) must be isomorphic to ℝ. The Recognition Science framework builds on this fact in a theorem called the bootstrap theorem, which addresses a foundational question: what number system is required for the framework's basic operation to work?

The framework's basic operation is a comparison operator, a function that takes two positive numbers and returns a third number. The framework models this operator with four conditions: identity (comparing a number with itself gives zero), non-contradiction (swapping the inputs gives the same result), scale invariance (scaling both inputs by a positive factor leaves the output unchanged), and distinguishability (some pair of inputs gives a nonzero output). These conditions are stated over a generic linearly ordered field, a number system where order and arithmetic behave as expected.

In Recognition Science, the comparison operator is central because it defines the cost of recognition, a measure of how much two recognized events differ. The framework's core result, the Law of Logic, derives a specific cost function from these conditions. But the Law of Logic requires a domain on which to operate. The bootstrap theorem closes this loop: it states that if a linearly ordered field supports the comparison operator and also satisfies the Archimedean property and Dedekind completeness, then that field is canonically isomorphic to ℝ. The Archimedean property means there are no infinite or infinitesimal elements; Dedekind completeness means every nonempty bounded set has a least upper bound.

The theorem is proved in the framework's machine-checked library of formal theorems. The proof reduces to the classical characterization of ℝ as the unique Archimedean Dedekind-complete ordered field. The theorem is named bootstrap_to_real and its corollary bootstrap_closure, both anchored in the module DomainBootstrap.lean. The framework's own recovered real line, called LogicReal, is isomorphic to ℝ, so the choice of ℝ as the domain is canonical up to isomorphism, not arbitrary.

What the bootstrap theorem does not claim is equally important. It does not derive the Archimedean completeness property from the Law of Logic alone. The Law of Logic on its own could operate on the rational numbers ℚ or any incomplete subfield. The completeness hypothesis is named explicitly as a residual analytic input, not hidden. The theorem also does not prove that ℝ is the only possible domain in an absolute sense; it proves uniqueness up to canonical isomorphism given the stated conditions. The framework's claim is that, with the standard analytic assumption, the real numbers are the unique domain that works.

/-- **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⟩

/-- **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 closure**: the Law of Logic plus Archimedean
completeness uniquely picks out `ℝ` as the ambient ordered field, up
to canonical isomorphism. -/
theorem bootstrap_closure
    (K : Type*) [ConditionallyCompleteLinearOrderedField K]
    (h : LogicSupported K) :
    Nonempty (K ≃+*o ℝ) :=
  bootstrap_to_real K h

What this page does not claim

The bootstrap theorem does not derive Archimedean completeness from the Law of Logic; it is a named hypothesis. The theorem does not prove uniqueness of ℝ without the completeness assumptions, since incomplete fields like ℚ also support the operator. The framework does not claim the comparison operator is the only possible structure on ℝ.

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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