Let be the class of recursively enumerable (r.e.) sets with infinite complements. A set M ϵ is maximal if every superset of M which is in is only finitely different from M. In [1] Friedberg shows that maximal sets exist, and it is an easy consequence of this fact that every non-simple set in has a maximal superset. The natural question which arises is whether or not this is also true for every simple set (Ullian [2]). In the present paper this question is answered negatively. However, the main concern of this paper is with demonstrating, and developing a few consequences of, what might be called the “density” of hyperhypersimple sets.

Since publication of our account of the early history of the Association for Symbolic Logic in the September 1962 issue of its Journal, the fact has been called to our attention that — because much of the correspondence at that time is no longer available and we had therefore to depend on our memory — we failed to mention an important item of that early history.

This was a letter from Professor Paul Weiss to the Editor of Philosophy of Science, which appeared in the April, 1934, issue of that periodical. This letter, entitled “A Home for Logic,” called attention to the fact that papers on logic were scattered in heterogeneous periodicals, and suggested that the Philosophy of Science Association take under its wing the establishment of a Logical Association.

Following Robinson [3], we say that a first-order sentence S is persistent with respect to a set of sentences K if, whenever S is true in a model M of K, then S is true in every extension of M that is also a model of K. Robinson proved in [3] that:

(T) In order that the sentence S be persistent with respect to the set K it is necessary and sufficient that there be some existential sentence Y such that the sentence S↔Y is deducible from K.