This paper continues the project initiated in [5]: a model-theoretic study of the concept of cardinality within certain higher-order logics. As recommended by an editor of this Journal, I will digress to say something about the project's motivation. Then I will review some of the basic definitions from [5]; for unexplained notation the reader should consult [5].

The syntax of ordinary usage (with respect to the construction of arguments as well as the construction of individual sentences) makes it natural to classify numerals and expressions of the form ‘the number of F's’ as singular terms, expressions like ‘is prime’ or ‘is divisible by’ as predicates of what Frege called “level one”, and expressions like ‘for some natural number’ as first-order quantifier-phrases. From this syntatic classification, it is a short step—so short as to be frequently unnoticed—to a semantic thesis: that such expressions play the same sort of semantic role as is played by the paradigmatic (and nonmathematical) members of these lexical classes. Thus expressions of the first sort are supposed to designate objects (in post-Fregean terms, entities of type 0), those of the second sort to be true or false of tuples of objects, and those of the third sort to quantify over objects. All this may be summed up in Frege's dictum: “Numbers are objects.”