Stationary Logic and Ordinals

D. G. Seese
1981 Transactions of the American Mathematical Society  
The L(aa)-theory of ordinals is investigated. It is proved that this theory is decidable and that each ordinal is finitely determinate. 0. Introduction. The quantifier "aaA"' with the intended meaning "there is a closed and unbounded system of countable sets X" was introduced by Shelah in [25] . If we enrich the logic Lau by this quantifier we get the stationary logic Lua(aa). The fundamental paper for the study of stationary logic is [1] . Stationary logic has a smooth model theory and is a
more » ... eralization of the logic Lwu(<2,) with the quantifier "there exist uncountably many". But unfortunately LUUJ(aa) does not possess all good properties of LUU(QX). For instance Luw(aa)-elementary equivalence is not preserved by performing disjoint unions and finite direct products. Moreover Harrington, Kunen and Shelah proved that the existence of L(aa)-elementary substructures of cardinality w, of all uncountable structures is independent of ZFC (given the consistency of the existence of a super compact cardinal). The reasons for this negative behaviour of stationary logic are the properties of the dual quantifier "-laaA'-i" denoted by "stat X". To avoid these difficulties Kaufmann introduced in his thesis [10] (see also [1] ) the notion of finitely determinate structures. A structure is said to be finitely determinate if it satisfies all instances of the scheme
doi:10.2307/1998646 fatcat:75up2yzha5f63iiiaxjjhl5woe