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This paper is about mathematical problems in programming language semantics and their influence on recursive function theory. We define a notion of computability on continuous higher types (for all types) and show its equivalence to effective operators. This result shows that our computable operators can model mathematically (i.e. extensionally) everything that can be done in an operational semantics. These new recursion theoretic concepts which are appropriate to semantics also allow us to construct Scott models for the &lgr;-calculus which contain all and only computable elements. Depending on the choice of the initial cpo, our general theory yields a theory for either strictly determinate or else arbitrary non-deterministic objects (parallelism). The formal theory is developed in part II of this paper. Part I gives motivation and comparison with related work.

Computability concepts for programming language semantics