In mathematics, primitive recursive set functions or primitive recursive ordinal functions are analogs of primitive recursive functions, defined for sets or ordinals rather than natural numbers. They were introduced by Jensen & Karp (1971).

Definition

A primitive recursive set function is a function from sets to sets that can be obtained from the following basic functions by repeatedly applying the following rules of substitution and recursion:

The basic functions are:

• Projection: P_n,m(x₁, ..., x_n) = x_m for 0 ≤ m ≤ n
• Zero: F(x) = 0
• Adjoining an element to a set: F(x, y) = x ∪ {y}
• Testing membership: C(x, y, u, v) = x if u ∈ v, and C(x, y, u, v) = y otherwise.

The rules for generating new functions by substitution are

• F(x, y) = G(x, H(x), y)
• F(x, y) = G(H(x), y)

where x and y are finite sequences of variables.

The rule for generating new functions by recursion is

• F(z, x) = G(∪_u ∈ z F(u, x), z, x)

A primitive recursive ordinal function is defined in the same way, except that the initial function F(x, y) = x ∪ {y} is replaced by F(x) = x ∪ {x} (the successor of x). The primitive recursive ordinal functions are the same as the primitive recursive set functions that map ordinals to ordinals.

Examples of primitive recursive set functions:

• TC, the function assigning to a set its transitive closure.^[1]: 26
• Given hereditarily finite ${\displaystyle c}$, the constant function ${\displaystyle f(x)=c}$. ^[1]: 28

Extensions

One can also add more initial functions to obtain a larger class of functions. For example, the ordinal function ${\displaystyle \alpha \mapsto \omega ^{\alpha }}$ is not primitive recursive, because the constant function with value ω (or any other infinite set) is not primitive recursive, so one might want to add this constant function to the initial functions.

The notion of a set function being primitive recursive in ω has the same definition as that of primitive recursion, except with ω as a parameter kept fixed, not altered by the primitive recursion schemata.

Examples of functions primitive recursive in ω:^{[1] pp.28--29}

• ${\displaystyle \mathbb {P} _{\omega }(x)=\bigcup _{n<\omega }x^{n}}$.
• The function assigning to ${\displaystyle \alpha }$ the ${\displaystyle \alpha }$th level ${\displaystyle L_{\alpha }}$ of Godel's constructible hierarchy.

Primitive recursive closure

Let ${\displaystyle f_{0}:{\textrm {Ord}}^{2}\to {\textrm {Ord}}}$ be the function ${\displaystyle f(\alpha ,\beta )=\alpha +\beta }$, and for all ${\displaystyle i<\omega }$, ${\displaystyle {\tilde {f}}_{i}(\alpha )=f_{i}(\alpha ,\alpha )}$ and ${\displaystyle f_{i+1}(\alpha ,\beta )=({\tilde {f}}_{i})^{\beta }(\alpha )}$. Let L_α denote the αth stage of Godel's constructible universe. L_α is closed under primitive recursive set functions iff α is closed under each ${\displaystyle f_{i}}$ for all ${\displaystyle i<\omega }$. ^[1]: 31

References

• Jensen, Ronald B.; Karp, Carol (1971), "Primitive recursive set functions", Axiomatic Set Theory, Proc. Sympos. Pure Math., vol. XIII, Part I, Providence, R.I.: Amer. Math. Soc., pp. 143–176, ISBN 9780821802458, MR 0281602