## Topological Incompleteness and Order Incompleteness of the Lambda Calculus (2001)

Venue: | ACM TRANSACTIONS ON COMPUTATIONAL LOGIC |

Citations: | 23 - 15 self |

### BibTeX

@ARTICLE{Salibra01topologicalincompleteness,

author = {Antonino Salibra},

title = {Topological Incompleteness and Order Incompleteness of the Lambda Calculus},

journal = {ACM TRANSACTIONS ON COMPUTATIONAL LOGIC},

year = {2001},

volume = {4},

pages = {2003}

}

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### OpenURL

### Abstract

A model of the untyped lambda calculus induces a lambda theory, i.e., a congruence relation on λ-terms closed under ff- and fi-conversion. A semantics (= class of models) of the lambda calculus is incomplete if there exists a lambda theory which is not induced by any model in the semantics. In this paper we introduce a new technique to prove the incompleteness of a wide range of lambda calculus semantics, including the strongly stable one, whose incompleteness had been conjectured by Bastonero-Gouy [6, 7] and by Berline [9]. The main results of the paper are a topological incompleteness theorem and an order incompleteness theorem. In the first one we show the incompleteness of the lambda calculus semantics given in terms of topological models whose topology satisfies a property of connectedness. In the second one we prove the incompleteness of the class of partially ordered models with finitely many connected components w.r.t. the Alexandroff topology. A further result of the paper is a proof of the completeness of the semantics of the lambda calculus given in terms of topological models whose topology is non-trivial and metrizable.