In [1], Sacks points out that there is one fundamental question: which true statements of ordinary recursion theory remain true when appropriately extended to metarecursion theory?

A particular interest is taken in the question [1]:

Q6. How does one define the jump operator for metarecursion theory? (A satisfactory definition should have the property that if A is metarecursive in B, then the jump of A is metarecursive in the jump of B.)

In [2], Kreisel and Sacks give some definitions of predicates and functions analogous to those of Kleene as follows:

The T-predicate of [2] is analogous to that of Kleene [3]. Its definition is

where e is the Gödel number of a finite system of equations E and t(e, s) is a special metarecursive function which indexes “deductions” from E.

U(e, s) is a metarecursive function such that if t(e, s) = 〈e, M, N, x, y〉, then U(e,s) = y.

Two partial functions {e} s and {e} are

Then e is intrinsically consistent if for all x, s 1 and s 2, if t(e, S 1) = 〈e, M 1 N 1, x, y 1〉, t(e, s 2) = 〈e, M 2, N 2, x, y 2〉 and (M 1 ∪ M 2) ∩ (N 1 ∪ N 2) = ∅, then y 1 = y 2.