A continuum of theories of lambda calculus without semantics

Abstract

In this paper, we give a topological proof of the following result: there exist 2¿(/spl aleph//sub 0/) lambda theories of the untyped lambda calculus without a model in any semantics based on D.S. Scott's (1972, 1981) view of models as partially ordered sets and of functions as monotonic functions. As a consequence of this result, we positively solve the conjecture, stated by O. Bastonero and X. Gouy (1999) and by C. Berline (2000), that the strongly stable semantics is incomplete.