Ordinal analysis of terms of finite type

Abstract

Sanchis [Sa] and Diller [Di] have introduced an interesting relation between terms for primitive recursive functionals of finite type in order to obtain a computability proof. The method is as follows. First the relation is shown to be well-founded. Then computability of each term is obtained by transfinite induction over the relation.Their relation is given by various clauses which define the (immediate) successors of a term. Hence, starting with a term H and repeatedly applying the successor relation one generates the tree of H (say). The problem is to prove that the trees are well-founded. In their proofs Sanchis and Diller use the axiom of bar induction. Diller also gives a proof using transfinite induction: each tree is shown to have a length b < φω(0) (the first ω-critical number) and the induction is over ordinals less than or equal to b.For proof-theoretic purposes it would be desirable to employ a method of proof which is more elementary than the axiom of bar induction and also to obtain a smaller ordinal bound. The purpose of the following is to provide such a method. We show that for a primitive recursive term H of finite type the tree generated by the relation of Sanchis and Diller has length less than ε0. Our method is similar to that used by Tait [Ta] in his theory of infinite terms. The immediately natural metamathematics is elementary intuitionistic analysis with the axiom of choice and transfinite induction over the relevant ordinal notations. We give several versions of the basic list of successor clauses, and for one version we show how to carry out the treatment in Skolem arithmetic (with primitive recursion of lowest type).

Representing definable functions of HA by neighbourhood functions

“…Specifically, given a closed term Y : N N → N of HA ω , one can construct a neighbourhood function of Y as a closed term of HA ω for which bar induction is valid. The existing literature suggests that our result is not surprising: it is known that a closed term Y : N N → N of system T has a T-definable modulus of continuity (see e.g., Schwichtenberg [13]); moreover, HA ω is closed under the rule of bar induction (Howard [6,Section 5]). 1 However, our proof does not rely on sophisticated proof theoretic methods such as normalisation of infinite terms or ordinal analysis used in those works.…”

Logic, Algebra, and Computation

“…However, using a technique due to (Howard 1980b) (which in tum is based on (Sanchis 1967) and (Diller 1968» it can be shown that we have the following superexponential universal bound. But it is not obvious how a reasonable estimate for that height might be obtained.…”

On the Cantor–Bendixson rank of a set that is searchable in Gödel’s T

“…In Section 3 we use a result by Howard [8] and show that when an operator of type 3 is T-definable, then it is computable via an evaluation tree of rank < ε 0 .…”