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A model of the untyped lambda calculus univocally induces a lambda theory (i.e., a congruence relation on λ-terms closed under α- and β-conversion) through the kernel congruence relation of the interpretation function. A semantics of lambda calculus is (equationally) incomplete if there exists a lambda theory that is not induced by any model in the semantics. In this article, we introduce a new technique to prove in a uniform way the incompleteness of all denotational semantics of lambda calculus that have been proposed so far, including the strongly stable one, whose incompleteness had been conjectured by Bastonero, Gouy and Berline. We apply this technique to prove the incompleteness of any semantics of lambda calculus given in terms of partially ordered models with a bottom element. This incompleteness removes the belief that partial orderings with a bottom element are intrinsic to models of the lambda calculus, and that the incompleteness of a semantics is only due to the richness of the structure of representable functions. Instead, the incompleteness is also due to the richness of the structure of lambda theories. Further results of the article are: (i) an incompleteness theorem for partially ordered models with finitely many connected components (= minimal upward and downward closed sets); (ii) an incompleteness theorem for topological models whose topology satisfies a suitable property of connectedness; (iii) a completeness theorem for topological models whose topology is non-trivial and metrizable.

Topological incompleteness and order incompleteness of the lambda calculus