# Gödel numberings, principal morphisms, combinatory algebras

## Abstract

Functional languages are based on the notion of application: programs may be applied to data or programs. By application one may define algebraic functions and a programming language is functionally complete when any algebraic function f(x_{1},...,x_{n}) is representable (i.e. there is a constant a such that f(x_{1},...,x_{n}) = ax_{1}· ... ·x_{n}). Combinatory Logic (C.L.) is the simplest type-free language which is functionally complete.

In a sound category-theoretic framework the constant a above may be considered an "abstract gödel-number" for f, as gödel-numberings are generalized to "principal morphisms". By this, models of C.L. are categorically characterized and their relation is given to λ-calculus models within Cartesian Closed Categories.

Finally, the partial recursive functionals in any finite higher type are shown to yield models of C.L..

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