DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-12

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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The zero, successor, and projection functions on the natural numbers

Definition

For each integer $k \geq 1$ , the initial arithmetic functions on $N$ are:

the zero function $Z : N \to N, Z (n) = 0;$
the successor function $S : N \to N, S (n) = σ (n) = n \cup {n};$
for each $1 \leq i \leq k$ , the $i$ th projection $P_{i}^{k} : N^{k} \to N, P_{i}^{k} (x_{1}, \dots, x_{k}) = x_{i} .$

Here $N$ is the natural-number system from The natural numbers $N$ (von Neumann), and each displayed rule determines a total function in the sense of A function is a relation $f$ with $(a, b) \in f$ and $(a, c) \in f$ implying $b = c$ ; $f : A \to B$ , the value $f (a)$ , domain and codomain.

Remarks

These are the basic generators from which primitive recursive functions are built.
The arity is part of the data: the family ${P_{i}^{k}}$ contains one projection for each positive arity $k$ and each coordinate $i$ in that arity.

Depends on

Used by

Composition and primitive recursion as closure schemes on arithmetic functions Definition
Primitive recursive functions Definition

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jeremy Avigad and Richard Zach, Recursive Functions (standard reference, not scraped)
Klaus Sutner, Primitive Recursion (standard reference, not scraped)