Sets of Formulas Valid in Finite Structures

A function is defined on the set of all subsets of so that for each set K, the value, , is the set of formulas valid in all structures of cardinality in K. An analysis is made of the dependence of on K, For any set K, let be the Kleene-Post degree to which K belongs. It is easily seen that for all infinite sets K, . On the other hand, we prove that , and use this to prove that, for any two degrees a and b, , and b r.e. a, there exists a set K so that and . Various similar results are also included.