Foundation Ordered Logic Realization Ordered Interpret Le Iff

The order preservation theorem

The natural numbers come with a familiar order: 0 is less than 1, 1 is less than 2, and so on. The theorem ordered_interpret_le_iff establishes that this order is preserved exactly when natural numbers are represented through a ledger, a discrete record of events with a forced cost. In plain terms, if one number is at most another in the usual sense, then the ledger-based representation also respects that ordering, and vice versa.

This is not a new ordering imposed on the numbers. The theorem shows the ledger representation and the standard order agree completely. The proof is machine-checked in the framework's library of formal theorems, meaning no step is left to hand-waving. The declaration sits inside a larger construction called OrderedLogicRealization, which builds a faithful arithmetic from the ledger concept.

What the theorem does not claim is broader. It does not say the ledger creates the natural numbers or their order from nothing. The order is taken as given on the natural numbers themselves; the theorem only confirms that the ledger-based view matches it. It also does not claim anything about other number systems, such as the integers or reals, whose orders behave differently. The result is specific to the natural numbers and their standard ordering.

The payoff is consistency. If the framework's ledger is to serve as a foundation for arithmetic, then it must not contradict the basic facts everyone already knows. This theorem is one check that it does not. A reader can now see that the ledger-based arithmetic, at least for the natural numbers, preserves the order that schoolchildren learn first.

/-- Order on the carrier matches the recovered Peano order. -/
theorem ordered_interpret_le_iff (a b : ArithmeticFromLogic.LogicNat) :
    ArithmeticFromLogic.LogicNat.toNat a ≤ ArithmeticFromLogic.LogicNat.toNat b ↔ a ≤ b := by
  exact (ArithmeticFromLogic.LogicNat.toNat_le a b).symm

/-- The ordered realization interprets arithmetic faithfully. -/
theorem ordered_faithful :
    LogicRealization.FaithfulArithmeticInterpretation natOrderedRealization where
  injective := by
    intro a b h
    exact (ArithmeticFromLogic.LogicNat.eq_iff_toNat_eq).mpr h
  zero_step_noncollapse := by
    intro n h
    have hnat := congrArg id h
    simp [natOrderedRealization] at hnat
    exact Nat.succ_ne_zero _ hnat.symm

What this page does not claim

The ledger creates the natural numbers or their order from nothing. The theorem applies to integers or reals, whose orders differ from the natural numbers. The theorem establishes any property of arithmetic beyond order preservation.

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

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