Foundation Arithmetic Of Canonical Peano Surface

The canonical arithmetic surface

The natural numbers are the counting numbers 0, 1, 2, 3, and so on, with two basic operations: a starting point (zero) and a way to move from one number to the next (the successor function, which turns 0 into 1, 1 into 2, and so forth). This structure is called a Peano algebra. The classical fact, proved by Giuseppe Peano in 1889, is that these two ingredients, together with the principle of mathematical induction, completely determine the behavior of the natural numbers. Any other set with a zero and a successor function that obeys the same rules is essentially the same set, just with different names for its elements.

In Recognition Science, the framework's machine-checked library of formal theorems proves a parallel statement. The declaration canonical_peanoSurface establishes that the natural numbers, built from the framework's logic, satisfy the three defining properties of a Peano algebra: zero is not the successor of any number, the successor function never maps two different numbers to the same result (it is injective), and mathematical induction holds. These three properties are exactly what make the natural numbers a well-founded structure on which arithmetic can be built.

The theorem goes further. It shows that this arithmetic object is initial, meaning it is the starting point from which all other Peano algebras can be reached by a unique structure-preserving map. This is the technical sense in which the natural numbers are canonical: given any other arithmetic structure that satisfies the same rules, there is exactly one way to translate the natural numbers into it. The framework's library proves this uniqueness for the natural numbers themselves, and also for any arithmetic object extracted from a general logic realization.

What the theorem does not claim is that the framework invents arithmetic from nothing. The natural numbers, their properties, and the principle of induction are classical mathematics, known and proved for over a century. The framework's contribution is to show that these structures arise from its own logical starting point, and that they are unique up to isomorphism. The declaration does not claim to derive the specific numerical values of constants, nor does it claim that the Peano axioms are the only way to formalize arithmetic. It establishes a correspondence between the framework's logic and the classical arithmetic that mathematicians already use.

/-- The Peano surface for the canonical arithmetic object. -/
theorem canonical_peanoSurface (R : LogicRealization) :
    PeanoSurface (canonical R) where
  zero_ne_step := by
    intro x h
    cases h
  step_injective := by
    intro a b h
    exact LogicNat.succ_injective h
  induction := by
    intro P h0 hstep n
    exact LogicNat.induction (motive := P) h0 hstep n

/-- `LogicNat` is initial among Peano objects. -/
def logicNat_initial : PeanoObject.IsInitial logicNatPeano where
  lift := logicNatLift
  uniq := by
    intro B f g
    rw [logicNatLift_unique_fun B f, logicNatLift_unique_fun B g]

What this page does not claim

The framework does not invent arithmetic from nothing; the Peano axioms are classical mathematics. The declaration does not claim to derive the specific numerical values of physical constants. The theorem does not claim that the Peano axioms are the only way to formalize arithmetic.

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

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