In recent years much effort has gone into the study of languages which strengthen the classical first-order predicate calculus in various ways. This effort has been motivated by the desire to find a language which is
(I) strong enough to express interesting properties not expressible by the classical language, but
(II) still simple enough to yield interesting general results. Languages investigated include second-order logic, weak second-order logic, ω-logic, languages with generalized quantifiers, and infinitary logic.
(Received June 23 1968)
1 This paper contains the principal results of the first half of the author's Ph.D. thesis , submitted to Stanford University in August, 1967. We wish to thank our thesis advisor, Professor Solomon Feferman, for the considerable time, advice, direction and encouragement which we received. We also thank Professors Georg Kreisel and Dana Scott, as well as Kenneth Kunen, for many interesting discussions and helpful suggestions.
2 This paper was written while the author was an N.S.F. Postdoctoral Fellow.