First order properties of pairs of cardinals

We consider models of a countable first order logic L with an identity symbol and predicate symbols U, Po, Pi , • • • , U being unary. A model % = {A, U%, Po$i> • • • ) for L is said to be a twocardinal model if A is infinite and the power of U% is less than the power of A. By a set of axioms for two-cardinal models we mean a set 2 of sentences of L such tha t §1 is a model of 2 if and only if there exists a two-cardinal model which is elementarily equivalent to St. Using results of Fuhrken [ l ] , Vaught [4] proved the following theorem.