Foundation Domain Bootstrap Identity On

The identity axiom

IdentityOn is a formal definition inside the Recognition Science framework. It states a property of a comparison operator, which is a function that takes two positive numbers and returns a third number. The property is simple: for any positive number x, comparing x with itself must return 0. In symbols, C x x = 0. This is the framework's version of the classical identity law, the idea that a thing is equal to itself.

The declaration is generic. It is not written only for the real numbers. It works over any field that has a zero element and an ordering, meaning any structure where you can say one number is less than another. This generality is deliberate. The framework wants to state its axioms in the widest possible setting, then see what those axioms force.

In Recognition Science, this axiom is one of four Aristotelian conditions that a comparison operator must satisfy. The other three are non-contradiction (C x y = C y x), scale invariance (C (λx) (λy) = C x y), and distinguishability (there exist x and y with C x y ≠ 0). Together with IdentityOn, these form the structure called LogicSupported. The framework's library proves that any field supporting this structure, if it is also Archimedean and Dedekind-complete, must be isomorphic to the real numbers. That theorem is bootstrap_to_real.

What IdentityOn does not claim is just as important. It does not say that the comparison operator is unique. Many different functions could satisfy C x x = 0. It does not say that the real numbers are the only possible domain; the axiom can be stated over the rationals or any ordered field. It does not say that the identity law alone forces anything about the structure of reality. It is a single axiom, one ingredient in a larger recipe.

The payoff of this axiom is that it helps close a chicken-and-egg problem. The framework derives a real line from its logic, but it needs a real line to state that logic. IdentityOn, as part of the LogicSupported structure, is what makes the bootstrap theorem work: the logic plus completeness forces the domain to be the reals, and the reals support the logic. The axiom is not the whole story, but it is a necessary first step.

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

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

/-- **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

IdentityOn does not assert that the comparison operator is unique. IdentityOn does not say that the real numbers are the only possible domain for the axiom. IdentityOn alone does not force any physical structure; it is one axiom among several.

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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