Morphisms and Partitions of V-sets

Abstract

The set theoretic connection between functions and partitions is not worthy of further remark. Nevertheless, this connection turns out to have deep consequences for the theory of the Ershov numbering of lambda terms and thus for the connection between lambda calculus and classical recursion theory. Under the traditional understanding of lambda terms as function definitions, there are morphisms of the Ershov numbering of lambda terms which are not definable. This appears to be a serious incompleteness in the lambda calculus. However, we believe, instead, that this indefinability is a defect in our understanding of the functional nature of lambda terms. Below, for a different notion of lambda definition, we shall prove a representation theorem (completeness theorem) for morphisms. This theorem is based on a construction which realizes certain partitions as collections of fibers of morphisms defined by lambda terms in the classical sense of definition.