Gödel numberings, principal morphisms, combinatory algebras
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(x1,...,xn) is representable (i.e. there is a constant a such that f(x1,...,xn) = ax1· ... ·xn). 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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