Foundation Integers From Logic Lt Relation Unique

The uniqueness of less-than

The integers are built from pairs of natural numbers, where the pair (a, b) stands for the difference a - b. Two pairs name the same integer when their cross-sums agree, so (3, 1) and (5, 3) both name 2. This is the classical Grothendieck construction, a standard way to invent negative numbers from nothing but addition and equality.

Once the integers exist, the framework asks which ordering relations can be defined on them. The answer, proved in its machine-checked library of formal theorems, is that only one relation can play the role of less-than. If a relation r agrees with the usual integer ordering on every pair of values, then r is exactly the framework's own less-than relation. The proof is short: the two relations agree point by point, so they are the same relation. The same argument works for the non-strict ordering, less-than-or-equal.

This uniqueness is a theorem about the framework's construction, not a claim about the physical world. It says nothing about how numbers behave in nature, and it does not derive the integers from scratch. The construction assumes natural numbers and addition already exist; it builds integers from those ingredients. The theorem's force is internal: within the framework, once you fix the usual ordering on the embedded integers, the ordering on the constructed integers is forced, with no room for a second choice.

The practical consequence is that the framework's integers behave exactly like the familiar ones. The less-than relation on the constructed objects is not an arbitrary choice but the unique relation consistent with the embedding. A reader who wants to know whether the framework's integers are the real integers can check this theorem: the ordering matches, the arithmetic operations match, and the algebraic laws hold. The construction is not a new kind of integer; it is a new path to the same object.

/-- Any strict relation with the required transport law is the canonical
pulled-back relation. -/
theorem lt_relation_unique
    (r : LogicInt → LogicInt → Prop)
    (h : ∀ a b, r a b ↔ toInt a < toInt b) :
    r = lt := by
  funext a b
  apply propext
  exact h a b

/-- Any non-strict relation with the required transport law is the canonical
pulled-back relation. -/
theorem le_relation_unique
    (r : LogicInt → LogicInt → Prop)
    (h : ∀ a b, r a b ↔ toInt a ≤ toInt b) :
    r = le := by
  funext a b
  apply propext
  exact h a b

What this page does not claim

This theorem does not derive the integers from nothing; it assumes natural numbers and addition already exist. It does not claim anything about how integers behave in the physical world. It does not prove that the framework's integers are the only possible integers, only that their ordering is unique.

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/IntegersFromLogic.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.

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