Primitive recursive set function

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

The basic functions are:

• Projection: P_n,m(x₁,...,x^{n) = x_m}
• Zero: F(x)=0
• Adding an element to a set: F(x,y) = x∪{y}
• Testing membership: C(x,y,u,v) = x if 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∈zF(u,x),z,x)

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{{citation}}: CS1 maint: extra punctuation (link)