Foundation Universal Forcing Strict Ordered

The strict ordered realization

In mathematics, a realization is a concrete object that satisfies a set of abstract axioms. The strict ordered realization is the simplest such object for Recognition Science's cost framework: it takes the integers as its universe and defines the cost of moving from one integer to another as 0 if they are the same, and 1 otherwise. This is a binary equality test, not a metric that measures distance. The cost between 3 and 5 is the same as between 3 and 100.

Two properties of this cost function are established. First, the cost of moving from an integer to itself is always zero. Second, the cost is symmetric: the cost from a to b equals the cost from b to a. These are the two axioms the framework requires of any recognition cost, and they are verified for this integer-valued function. The proofs are direct, relying only on the definition and case analysis.

The construction then shows that this strict ordered realization carries an arithmetic structure equivalent to the natural numbers. It builds a Peano carrier and provides an equivalence to the standard logic-based natural numbers. This means the integers with this cost function support the same arithmetic as the counting numbers, even though the underlying set is larger and the cost is trivial.

In Recognition Science, the cost function is the primitive object from which structure is forced. This construction demonstrates that the framework's axioms admit at least one concrete model on the integers, with the simplest possible cost: equality or difference. It establishes that the axioms are consistent with a discrete, ordered universe, and that such a universe can still support arithmetic. The result is a baseline: if the framework can force structure from this minimal cost, it can force structure from richer ones.

@[simp] theorem intCost_self (a : ℤ) : intCost a a = 0 := by
  simp [intCost]

theorem intCost_symm (a b : ℤ) : intCost a b = intCost b a := by
  by_cases h : a = b
  · subst h
    simp [intCost]
  · have h' : b ≠ a := by intro hb; exact h hb.symm
    simp [intCost, h, h']

def strictOrdered_arith_equiv_logicNat :
    (StrictLogicRealization.arith strictOrderedRealization).peano.carrier
      ≃ ArithmeticFromLogic.LogicNat :=
  (StrictLogicRealization.toLightweight strictOrderedRealization).orbitEquivLogicNat

What this page does not claim

This construction does not derive the golden ratio or any specific physical constant. The strict ordered realization is not claimed to be the unique model of the cost axioms. The equivalence to natural numbers does not imply the integers are the preferred universe for the framework.

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/UniversalForcing/Strict/Ordered.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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