Assignment of Ordinals to Terms for Primitive Recursive Functionals of Finite Type

Abstract

Publisher Summary This chapter discusses the assignment of ordinals to terms for primitive recursive functionals of finite type. Gentzen showed that the consistency of first order arithmetic can be proved by methods that are finitistic except for the use of the descending chain principle for the ordinals less than ɛ 0 . The chapter extends its results to the case of arbitrary reductions by the use of non-unique assignments of ordinals to terms. The theory of expressions has interpretations in which the constants are ordinals and the variables range over a set of ordinals (or ordinal notations). In such interpretations an expression is interpreted as a function of the variables that it contains. By the expression assigned to H means that the initial component h 0 of the vector h assigned to H . When expressions are interpreted by means of ordinals the expression h 0 becomes an ordinal if H is closed, and this is taken to be the ordinal assigned to H .