Type Theories, Normal Forms and D_\infty-Lambda-Models

Abstract A (non-standard) inverse-limit λ-model D ∞ ∗ is constructed which has a non Hilbert-Post complete theory. Moreover, in D ∞ ∗ , a simple semantic characterization of normalizable terms is given. These results are proved using the properties of a generalized type assignment system which yields a filter model (Barendregt, Coppo, and Dezani-Ciancaglini, 1983 , J. Symbolic. Logic, 48, 931–940; Coppo, Dezani-Ciancaglini, Honsell, and Longo, 1983 , pp. 241–262, “Logic Colloquium '82,” North-Holland, Amsterdam), isomorphic to D ∞ ∗ . The type assignment system is also proved complete with respect to an interpretation of types (in the term model of β-equality) based only on normalization properties. As an application a class of maximal monoids of normalizable terms is characterized.

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