The Visser topology of lambda calculus

A longstanding open problem in lambda calculus is whether there exists a non-syntactical model of the untyped lambda calculus whose theory is exactly the least λ-theory λβ. In this paper we make use of the Visser topology for investigating the related question of whether the equational theory of a model can be recursively enumerable (r.e. for brevity). We introduce the notion of an effective model of lambda calculus and prove the following results: (i) The equational theory of an effective model cannot be λβ, λβη; (ii) The order theory of an effective model cannot be r.e.; (iii) No effective model living in the stable or strongly stable semantics has an r.e. equational theory. Concerning Scott’s semantics, we investigate the class of graph models and prove the following, where “graph theory” is a shortcut for “theory of a graph model”: (iv) There exists a minimum order graph theory (for equational graph theories this was proved in [9, 10]). (v) The minimum equational/order graph theory is the theory of an effective graph model. (vi) No order graph theory can be r.e. (vii) Every equational/order graph theory is the theory of a graph model having a countable web. This last result proves that the class of graph models enjoys a kind of (downwards) Löwenheim-Skolem theorem, and it answers positively Question 3 in [4, Section 6.3] for the class of graph models.