Foundation Domain Bootstrap Non Contradiction On

NonContradictionOn

In mathematics and physics, a comparison operator is a function that takes two numbers and returns a third, such as a cost or a distance. The declaration NonContradictionOn formalizes a symmetry property for such an operator: for any two positive numbers x and y, the result of comparing x with y must equal the result of comparing y with x. In symbols, the condition is C x y = C y x for all positive x and y. This is a reciprocity requirement, ensuring the comparison does not secretly depend on which argument comes first.

This condition is one of four Aristotelian conditions that Recognition Science uses to define a recognition event, a discrete act of comparing two quantities. The other conditions are identity (comparing a number with itself gives zero), scale invariance (scaling both inputs leaves the result unchanged), and distinguishability (some comparison is nonzero). Together, these conditions are packaged into a structure called LogicSupported, which requires a field with an order and a comparison operator satisfying all four properties.

The central theorem involving NonContradictionOn is the bootstrap theorem. It states that if a linearly ordered field K supports such a comparison operator, and if K is also Archimedean and Dedekind-complete, then K is canonically isomorphic to the real numbers ℝ as an ordered field. The proof reduces to the standard mathematical fact that ℝ is the unique Archimedean Dedekind-complete ordered field. This theorem is proved in the machine-checked library of formal theorems, with the declaration bootstrap_to_real.

In Recognition Science, this result closes a chicken-and-egg problem. The framework's Law of Logic is stated using a comparison operator on ℝ, but the framework also derives its own real line from the same Law. The bootstrap theorem shows that any other ambient field satisfying the same conditions must be isomorphic to ℝ, so the choice of ℝ as the domain is canonical up to isomorphism, not contingent. The theorem also makes explicit that the Law of Logic alone does not force Archimedean completeness; that completeness is a named, residual analytic input.

NonContradictionOn does not claim that the Law of Logic alone determines the real numbers. It requires the additional hypotheses of Archimedean and Dedekind completeness. It also does not claim that any comparison operator satisfying the four conditions is unique; the theorem only establishes an isomorphism between the ambient field and ℝ, not a unique operator. The declaration itself is a definition, not a theorem, so its content is a formal specification of a symmetry condition, not a derived result.

/-- Non-contradiction, generic field version. -/
def NonContradictionOn [LT K] [Zero K] (C : ComparisonOperatorOn K) : Prop :=
  ∀ x y : K, 0 < x → 0 < y → C x y = C y x

/-- **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

NonContradictionOn alone forces the ambient field to be the real numbers. The comparison operator satisfying the four conditions is unique. The bootstrap theorem provides a way to construct the real numbers from nothing.

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)

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