Foundation Ordered Logic Realization Ordered Faithful

Faithful arithmetic

The natural numbers are the counting numbers: 0, 1, 2, and so on, with the rule that every number has a distinct successor. A recognition ledger, a discrete record of events where comparing two entries costs either zero or one, can be built from these numbers in a simple way. The question is whether that built-in arithmetic is the real thing or a lookalike. The framework's machine-checked library of formal theorems proves it is the real thing: the theorem ordered_faithful shows that the arithmetic recovered from this ledger is a faithful interpretation of the natural numbers, meaning distinct numbers stay distinct and zero is not the successor of any number. In plain terms, the counting numbers you learned in school are exactly the numbers the ledger produces, with no extra identifications and no collapsed structure.

The proof works by defining a cost function on natural numbers: the cost of comparing two numbers is 0 if they are equal and 1 otherwise. This cost is symmetric, and equal numbers cost nothing. From this cost, the framework recovers a Peano arithmetic, the standard axiomatization of the natural numbers. The theorem then verifies two essential properties. First, the recovered arithmetic is injective: two different numbers in the ledger remain different in the recovered arithmetic. Second, zero does not collapse onto a successor: the zero of the ledger is not the successor of any number, so the counting never loops back on itself. Together these properties guarantee the recovered arithmetic has the same shape as the ordinary natural numbers, a result the library states as an equivalence between the recovered carrier and the standard one.

This matters because the framework builds larger structures, including logic and forcing, on top of this arithmetic. If the recovered numbers were a distorted copy, everything above them would inherit the distortion. The theorem removes that worry at the base: the foundation is exactly the counting numbers, no more and no less. It is a structural guarantee, not a claim about what numbers are in some metaphysical sense, but about how the ledger's arithmetic lines up with the familiar one.

What the theorem does not claim is just as important. It does not claim that this particular ledger is the only possible one, nor that the natural numbers are somehow created by the framework. The theorem shows a faithful interpretation exists for this construction, not that no other arithmetics exist. It also does not claim anything about the physical world; it is a purely formal result about the relationship between a defined cost function and the standard natural numbers. The framework's larger ambitions, such as deriving physical constants, stand on separate theorems, not on this one.

/-- 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

/-- Equality cost on `Nat`. -/
def natCost (m n : Nat) : Nat :=
  if m = n then 0 else 1

/-- 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

/-- 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 theorem does not claim this is the only possible arithmetic realization. The theorem does not claim the natural numbers are created by the framework. The theorem does not make any physical or empirical claim.

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)

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: