On predicates with constructive infinitely long expressions

Abstract

In recent years the logic with infinitely long expressions has been considered and developed by Berkeley school. ( For this see [6].) In this paper, we shall consider ' constructive ' infinitely long expres- sions. In the following we shall give a language for the logic with infinitely long expressions and define a ' formula with constructive infinitely long expressions' as a formula with infinitely long expressions (sometimes called simply a formula) to which a so-called Godel number is assigned. We shall show that the nesting number of a formula with constructive infinitely long expressions (see below) is less than Church-Kleene's $\omega_{1}$ (Theorem 1). Moreover we shall establish a correspondence between formulas with constructive infinitely long expressions and predicates in Kleene's analytic hierarchy (cf.