An accessibility proof of ordinal diagrams in intuitionistic theories for iterated inductive definitions

Abstract

Let (/, -<) be a non-empty well-ordered system with the least element 0, and / be /W{co} with the largest element oo. Let A be a non-empty well-ordered set. Then 0(1, A) denotes the system of ordinal diagrams (o.d.'s) based on / and A. (cf. [9, §26].) The accessibility proof for 0(1, A) in [9, pp. 298-309] shows that every o.d. from 0(1, A) is accessible with respect to <t for every i in /. The central notions in this proof are /-fans and /-accessibility for i in /. Roughly speaking, an o.d. ft is an /-fan if for every /-</ and every /-section v of fi,v is /-accessible, and an o.d. is /-accessible if it is accessible in /-fans with respect to <*. Consider the case when the order type of (/, -<) is a successor ordinal £+1. If we formalize this accessibility proof for O(£+l, 1) (=0(1, 1)) naturally, then this proof can be done in the intuitionistic theory lD\+i for £+l-times iterated inductive definitions. The purpose of this paper is to show the following fact: the accessibility of each o.d. from 0(6+1, 1) with respect to <0 is derivable in DDL (Theorem)