Foundation Domain Bootstrap Logic Supported

The bootstrap result

The real numbers are the complete ordered field: the number system where every bounded increasing sequence has a limit. A classical theorem, proved in the late 1800s by Dedekind and Cantor, says this property is unique. Any other structure with the same completeness and order properties is the real numbers in disguise, connected by a structure-preserving relabeling called an order isomorphism.

Recognition Science builds its framework on a comparison operator: a function that takes two positive quantities and returns a cost-like value, with the four Aristotelian conditions of identity, non-contradiction, scale invariance, and distinguishability. The framework's ledger, a discrete record of recognition events, uses this operator throughout. The question arises: could the whole theory run on some other number system, say the rationals or a p-adic field, instead of the reals?

The answer is no, provided one natural condition holds. The declaration LogicSupported packages the requirements: a linearly ordered field with a comparison operator satisfying the four conditions, plus the additional property that the field is Archimedean (no infinitely large or small elements) and conditionally complete (every bounded set has a least upper bound). The bootstrap theorem proves that any such field is canonically isomorphic to the real numbers as an ordered field. The proof reduces directly to the classical characterization of the reals, using the standard Mathlib result that an Archimedean conditionally complete linearly ordered field is order-isomorphic to ℝ.

The theorem closes a chicken-and-egg loop in the framework. The framework starts with ℝ as its ambient field, then recovers an isomorphic copy of it internally as LogicReal. The bootstrap theorem runs the other way: any ambient field that can support the framework's comparison operator must itself be ℝ, up to canonical isomorphism. The choice of ℝ as the domain is therefore not contingent but forced, given the analytic completeness assumption. The theorem also shows idempotence: ℝ itself is Logic-supported, and applying the theorem to ℝ yields nothing new.

What the theorem does not claim is equally important. The Archimedean and completeness hypotheses are named explicitly as residual analytic inputs; the Law of Logic on its own does not single out ℝ. Without those hypotheses, the comparison operator could live on the rationals ℚ or any incomplete subfield. The framework names this input rather than hiding it. The theorem also does not construct the comparison operator from nothing: it assumes one exists and proves the ambient field must be ℝ, not that recognition events force the reals into existence.

/-- **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

/-- **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⟩

/-- **Idempotence**: `ℝ` itself is a Logic-supported domain (witnessed
by any of the comparison operators we already have over `ℝ`). The
bootstrap theorem then says nothing new on `ℝ`, but on any other
candidate ordered field it forces an isomorphism to `ℝ`. -/
def real_supports_logic
    (C : LogicAsFunctionalEquation.ComparisonOperator)
    (h : LogicAsFunctionalEquation.SatisfiesLawsOfLogic C) :
    LogicSupported ℝ where
  zero_lt_one_in_K := by norm_num
  C := C
  identity := h.identity
  non_contradiction := h.non_contradiction
  scale_invariant := h.scale_invariant
  distinguishability :=
    LogicAsFunctionalEquation.distinguishability_of_nonTrivial C h.non_trivial

What this page does not claim

The Law of Logic alone forces the ambient field to be the reals without any completeness assumption. The theorem constructs the comparison operator or derives its existence from recognition events. The framework proves the real numbers exist; it takes them as an ambient ordered field and proves uniqueness up to isomorphism.

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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