In [6, p. 317] Curry described a formal system assigning types to terms of the type-free λ-calculus. In  Scott gave a natural semantics for this type assignment and asked whether a completeness result holds.
Inspired by  and  we extend the syntax and semantics of the Curry types in such a way that filters in the resulting type structure form a domain in the sense of Scott . We will show that it is possible to turn the domain of types into a λ-model, among other reasons because all λ-terms possess a type. This model gives the completeness result for the extended system. By a conservativity result the completeness for Curry's system follows.
Independently Hindley ,  has proved both completeness results using term models. His method of proof is in some sense dual to ours.
For λ-calculus notation see .
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