Delta Kernel Syntax Of Nat

The numeral map

In formal logic, a numeral is a name for a natural number built from the language's own symbols. The declaration ofNat in the δ-Kernel defines exactly this: it maps each ordinary natural number to a term in the framework's object language. The map sends 0 to the constant zero, and sends n + 1 to the successor of the image of n. So the numeral for 3 is succ (succ (succ zero)), a term built by applying the successor symbol three times to the zero symbol.

This definition is deliberately spare. It uses only the two primitive symbols of the distinction signature, zero and succ, which the framework treats as the basic building blocks of its discrete record of events. The map is a structural recursion: it pattern-matches on the input natural number and builds the corresponding term without any choice, classical logic, or appeal to higher-order machinery. The definition is total, meaning it produces a term for every natural number, and it is the canonical way the framework's object language refers to the metatheoretic naturals.

In Recognition Science, this numeral map is part of the syntax layer of the δ-Kernel, the framework's formal object logic. The object logic is intuitionistic first-order arithmetic over the signature {0, S, +, ·}, with equality as its sole atomic predicate. The declaration establishes that the object language can name every natural number, and that these names are built in a uniform, recursive way from the primitive symbols. It does not, by itself, assert that any arithmetic statement is true; it only provides the vocabulary in which such statements can be written.

The map also carries a specific philosophical weight within the framework. Because the object logic deliberately has no universe hierarchy, no Π-types, no inductive-type scheme, no propositions-as-types, no membership, and no comprehension, the numeral map is one of the few ways the framework connects its formal language to the ordinary counting numbers. The framework describes this as the "canonical image of the metatheoretic naturals," meaning it is the standard, forced way to translate a natural number into the object language, given the primitive signature.

What the declaration does not claim is equally important. It does not define addition or multiplication; those are separate, recursion-licensed extensions with their own defining equations. It does not prove any property of the natural numbers, such as commutativity or associativity. It does not assert that the object language is complete or that it can express every mathematical statement. It is purely a syntactic map, a bridge from the metatheory's naturals to the object language's terms, with no semantic content attached.

/-- Numerals: the canonical image of the metatheoretic naturals. -/
def ofNat : Nat → DTerm
  | 0     => zero
  | n + 1 => succ (ofNat n)

/-- Formulas of intuitionistic first-order arithmetic over the distinction
signature. Equality is the sole atomic predicate (identity of ledger
content). Negation is defined: `¬φ := φ → ⊥`. -/
inductive DFormula : Type where
  | eq   : DTerm → DTerm → DFormula
  | fls  : DFormula
  | conj : DFormula → DFormula → DFormula
  | disj : DFormula → DFormula → DFormula
  | impl : DFormula → DFormula → DFormula
  | all  : DFormula → DFormula
  | ex   : DFormula → DFormula

/-- Numerals: the canonical image of the metatheoretic naturals. -/
def ofNat : Nat → DTerm
  | 0     => zero
  | n + 1 => succ (ofNat n)

What this page does not claim

The declaration does not define addition or multiplication. The declaration does not prove any arithmetic property such as commutativity. The declaration does not assert semantic completeness of the object language.

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

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