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, s1 and s2, if t(e, S1) = 〈e, M1N1, x, y1〉, t(e, s2) = 〈e, M2, N2, x, y2〉 and (M1 ∪ M2) ∩ (N1 ∪ N2) = ∅, then y1 = y2.