Towards Lambda Calculus Order-Incompleteness

Abstract

After Scott, mathematical models of the type-free lambda calculus are constructed by order theoretic methods and classified into semantics according to the nature of their representable functions. Selinger [48] asked if there is a lambda theory that is not induced by any non-trivially partially ordered model (order-incompleteness problem). In terms of Alexandroff topology (the strongest topology whose specialization order is the order of the considered model) the problem of order-incompleteness can be also characterized as follows: a lambda theory T is order-incomplete if, and only if, every partially ordered model of T is partitioned by the Alexandroff topology in an infinite number of connected components (= minimal upper and lower sets), each one containing exactly one element of the model. Towards an answer to the order-incompleteness problem, we give a topological proof of the following result: there exists a lambda theory whose partially ordered models are partitioned by the Alexandroff topology in an infinite number of connected components, each one containing at most one λ-term denotation. This result implies the incompleteness of every semantics of lambda calculus given in terms of partially ordered models whose Alexandroff topology has a finite number of connected components (e.g.the Alexandroff topology of the models of the continuous, stable and strongly stable semantics is connected).