Easiness in graph models

Abstract

We generalize Baeten and Boerboom's method of forcing to show that there is a fixed sequence ( u k ) k ∈ ω of closed (untyped) λ -terms satisfying the following properties: (a) For any countable sequence ( g k ) k ∈ ω of Scott continuous functions (of arbitrary arity) on the power set of an arbitrary countable set, there is a graph model such that ( λ x . xx ) ( λ x . xx ) u k represents g k in the model. (b) For any countable sequence ( t k ) k ∈ ω of closed λ -terms there is a graph model that satisfies ( λ x . xx ) ( λ x . xx ) u k = t k for all k . We apply these two results, which are corollaries of a unique theorem, to prove the existence of (1) a finitely axiomatized λ -theory L such that the interval lattice constituted by the λ -theories extending L is distributive; (2) a continuum of pairwise inconsistent graph theories (= λ -theories that can be realized as theories of graph models); (3) a congruence distributive variety of combinatory algebras (lambda abstraction algebras, respectively).