Axiomatization of the monadic second order theory of ω1

Abstract

For any ordinal ~ , let MT [~] be the Monadic second order Theory of ~ , i.e. the theory where quantification over both, ordinals less than ~ and subsets of ~ , is allowed and the order relation is the only primitive. Let MT ~o] be the Monadic second order Theory of Countable Ordinals, i.e. the intersection of all MT [~] for ~<~I MT[~] for ~<~ and MT[co~ are decidable by B~chi [BG 4] . The d e c i d a b i l i t y of MT[~I] i s proved in the preceding paper [B~] , where also detailed proofs for the results of [Bi 4~ are given. In this paper we characterize MT ~o] and MT [~] for ~ ~d by systems. A further paper [BS q] on the subject is planned: There all consistent and complete extensions of HT[oo] (which include MT[~] for ~ <~d) will be characterized by systems; it will be proved that the Boolean algebra of sentences of HT[oo] has an ordered basis of type ~+2 -q. In the interpretation of an MS-formula the set quantifiefs range over all subsets of the domain~ in general, in an interpretation the set quantifiers may be restricted to a certain family of subsets of the domain. A set ~ of sentences is a (standar@axiom for the MS-theory 7 ~ iff ~ is recursive and exactly the true sentences of ~ are the (standard) consequences of ~ , i.e. hold in all (standard) models of ~. Caution: What we call standard system here, is called simply axiom system in [Bd] . Actually, for ~ ~ ~, the systems for HT [~] which are easily drawn from [Bd] , are systems; for ~=~ this was shown in Siefkes [Si ~ . For ~ ~ ~, there are certain consequences of the of choice which are independent from the axioms for NT[~] , i.e. which need not hold in (general) models of the