Ordinal Trees and Computable Ordinals

Abstract

In chapter 2.6 we have indicated how appropriate finite path trees can be used for specifying infinite computable production processes. We shall generalize Definition 2.7.5 of trees by also admitting nodes with infinite branching. Such nodes will serve as names for functions with infinitely many arguments such as “$\mathop {\lim }\limits_{n \to \infty }$”. For measuring the height of such trees the denumerable ordinal numbers, i.e. the numbers of Cantor’s second number class, are needed. We shall define the effective numberings of the computable trees over the (unrestricted) signature a. As the main application we derive a standard numbering of the computable ordinals. We prove that the computable ordinals can be characterized by total numberings of well-orders for which v(i) ≤ v(j) is decidable. Our approach differs slightly from Kleene’s original one.