Accessible recursive functions

The class of all recursive functions fails to possess a natural hierarchical structure, generated predicatively from “within”. On the other hand, many (proof-theoretically significant) sub-recursive classes do. This paper attempts to measure the limit of predicative generation in this context, by classifying and characterizing those (predictably terminating) recursive functions which can be successively defined according to an autonomy condition of the form: allow recursions only over well-orderings which have already been “coded” at previous levels. The question is: how can a recursion code a well-ordering? The answer lies in Girard’s theory of dilators, but is reworked here in a quite different and simplified framework specific to our purpose. The “accessible” recursive functions thus generated turn out to be those provably recursive in (Π1 − CA)0. Introduction. Before one accepts a computable function as being recursive, a proof of totality is required. This will generally be an induction over the tree of sub-computations unravelled from the defining algorithm, since termination corresponds to well-foundedness along the computationbranches. If the tree is well-founded, the strength of the induction principle over its Kleene-Brouwer well-ordering thus serves as a measure of the prooftheoretical complexity of the given function. The aimof this paper is to isolate and characterize those recursive functions whichmay be termed “predictably terminating” or “predicatively accessible” according to the following hierarchy principle: one is allowed to generate a function at a new level only if it is provably recursive over a well-ordering already coded in a previous level, i.e., only if one has already constructed a method to prove its termination. This begs the question: what should it mean for a well-ordering to be “coded in a previous level”? Certainly it is not enough merely to require that the characteristic function of its ordering relation should have been generated at an earlier stage, since the resulting hierarchywould then collapse in the sense that all recursive functions would appear immediately once the ∆0 relations had been produced. This is simply because every recursive Received September 30, 1998; revised April 16, 1999. Lecture given at the Association for Symbolic Logic Annual Meeting in Toronto, May 1998, and at the Symposium in Honour of S. Feferman held at Stanford, December 1998. c © 1999, Association for Symbolic Logic 1079-8986/99/0503-0004/$3.20