In this paper it is shown that, if a certain form of Gödel-Bernays set theory which does not include the axiom of extensionality is consistent, then so is the whole system of set theory. The general line of argument is similar to that used in Part I. § 1 describes a reformulation of set theory in which the class-existence axioms are replaced by the use of abstracts. In § 2 some standard theorems, including the theory of the ancestral, are proved without using the axiom of extensionality. In § 3 the appropriate inner model is defined, and the validity in it of the most of the axioms is demonstrated. § 4 deals with the remaining axioms (of infinity and of choice).

In this note we present an axiomatization of the classical two-valued propositional calculus, for which proofs of decidability, consistency, completeness, and independence, are almost trivial (given an understanding of truth tables).

Venn constructed diagrams of up to five simply connected regions that overlapped each other once in each possible way of overlapping. Although Venn did not prove that his diagrams were constructible for more than five simply connected regions — in fact, he preferred to have a doubly connected region in his 5-class diagram — he summarized his method of construction with an intuitive argument: “But for merely theoretical purposes the rule of formulation would be very simple. It would merely be to begin by drawing any closed figure, and then proceed to draw others, subject to the one condition that each is to intersect once and once only all the existing subdivisions produced by those which had gone before.” The method of construction given below leads to a simple topological proof that Venn diagrams can be constructed for any number of simply connected regions.

Independence of axioms was not investigated in [Faris]. We show that this quality is lacking, since 2 can be derived from the rest, which without it constitute an independent set. In the proofs of independence it is clearly more informative to decompose 9, C1abKN2abKN3abN5ab and 10, C3abKN2abN5ab into three and two propositions respectively.

Horn [2] obtained a sufficient condition for an elementary class to be closed under direct product. Chang and Morel [1] showed that this is not a necessary condition. We will show that, if consideration is restricted to identity theory, that is, a first-order predicate calculus with equality but no other relation symbols or operation symbols, Horn's condition is necessary and sufficient.

A model for identity theory consists of a non-empty domain A, but no relations or operations except equality. If I is an index set, and for each is a model for identity theory, then the direct product of the is a model for identity theory and has domain A, the cartesian product of the Ai.

It is known [1] p. 312 that a simultaneous enumeration of the recursively enumerable sets can be achieved by a Σ10-predicate. We shall show that a simultaneous enumeration of the recursive sets can be achieved by a predicate of the same form.