Feedback Hyperjump

This child is to be thought of as a piece of syntax acting as a placeholder, and not, for instance, as feedback computation, for which angle brackets \(\langle \rangle \) are also used.

Sometimes we will have occasion to consider the computation \(\{\bar{e}\}(k)\) instead. Then implicity \(e = \langle \bar{e}, k \rangle \).

It bears mention that there are several options for dealing with this clause. In all cases, the evidence that n is not an ordinal notation is that its tree \(U^{P}_{n}\) of smaller ordinal notations is bad somehow, either ill-formed or ill-founded. For \(P^+\), we took this in the strictest possible sense: \(U^{P}_{n}\) had to be non-freezing in order to qualify as evidence. For \( P^{ \& }\), there is no such requirement on \(U^P_n\) ever; once we have any evidence that \(U^{P}_{n}\) will not be acceptable, we take it. In contrast with both of these, one could work in the middle. That is, the reasons that \(U^{P}_{n}\) activate clause 2 are that it has a node not of the right form, or that the function named by a node is partial, or that the function named by a node is not increasing (in the sense of \(<_{P}\)), or that the tree has an infinite descending path; the requirement that \(U^{P}_{n}\) be non-freezing could, in principle, be levied on some and not all of these conditions. We find the two extreme cases isolated here to be the most natural ones; we believe that the only condition of any real importance is the well-foundedness of \(U^{P}_{n}\), and that varying the others will make no difference; determining this is left for future work.