Recursively Saturated Models of Set Theory

We determine when a model 90? of ZF can be expanded to a model of a weak set-and-class theory-stronger than GB (Gödel-Bernays) but much weaker than MK (Morse-Kelley). Thus 9C will be a collection of classes-a collection of subsets of 3ft. We shall consider <3ft, 9C > to be a Henkin model for second order logic. Thus second order (capital letter) variables vary over classes-elements of 9C-and first order (small letter) ones, over sets-elements of 3ft. A formula is said to be first order if it contains no class quantifiers-it may have class parameters. So a 2¡ formula is a formula of the form 3X by both 2] and n¡ formulas. We say <9ft, % > 1= A{ CA (Aj comprehension axiom) if every A¡ definable subset of 3ft is an element of %. (This can be expressed with an obvious axiom scheme.) We ask when 3ft N ZF can be expanded to a model <3ft, 9C > t= GB + Aj CA. That the problem is not trivial is suggested by the following two well-known results: If 3ft N ZF and % is the collection of definable subsets of 3ft, then <3ft, 9C> N GB. And if <3ft, %> N MK, 3ft N Con(ZF). (For this we need only that Received by the editors March 31, 1978 and, in revised form, July 16, 1979; principal results were presented at the annual meeting of the Association for Symbolic Logic in St. Louis, Missouri, January 28, 1977. AMS (MOS) subject classifications (1970). Primary 02B15, 02F27, 02H20, 02K99; Secondary 02B25, 02H10, 02K20.