Wilfrid Hodges

Elementary Predicate Logic

Elementary (first-order) predicate logic is a child of many parents. At least three different groups of thinkers played their part in its conception, with three quite distinct motives. Maybe the mixture gave it hybrid strength. But whatever the reason, first-order logic is both the simplest, the most powerful and the most applicable branch of modern logic.

Interpreting number theory in nilpotent groups

Corollary 1. For each k >_ 2, there are continuum many elementary equivalence types of existentially closed nilpotent groups of class k. Corollary 2 (Saracino [4]). For each k >= 2, the theory of niIpotent groups of class k has no model companion. Proof. The variety of nilpotent groups of class k has the joint embedding property, so that if there was a model companion, all existentially closed models would be elementary equivalent (cf. Hirschfeld and Wheeler [1], Theorem 2.4, Proposition 2.8), contradicting Corollary 1. []

Constructing pure injective hulls

Let A be an abelian group and B a pure injective pure extension of A . Then there is a homomorphic image C of B over A which is a pure injective hull of A ; C can be constructed by using Zorns lemma to find a suitable congruence on B . In a paper [4] which greatly generalises this and related facts about pure injectives, Walter Taylor asks (Problem 1.5) whether one can find a “construction” of C which is more concrete than the one mentioned above; he asks also whether the points of C can be explicitly described. In this note I return the answer No.