The Complete Extensions of the Monadic Second Order Theory of Countable Ordinals

Abstract

Let MT[co] be the Monadic second order Theory of Countable Ordinals. Thus MT[co] has individual variables and set variables, both quantifiable. The only primitive is the symbol < for the order relation. Call this language ℒ. The true sentences of MT[co] are those of ℒ which hold in all countable ordinals. Here “countable ordinal” refers to models of set theory, say Zermelo-Fraenkel, plus the axiom of choice. (For details compare the appendix, Section 7). From MT[co] we get ET[co], the Elementary Theory of Countable Ordinals, by cancelling the set variables. Let ℒ 0 be the language of ET[co]. Occasionally we will consider the Weak monadic second order Theory of Countable Ordinals, WT[co]. which has the same language as MT[co], but the set variables are interpreted as ranging over finite sets only. Both ET[co] and WT[co] are the same as the corresponding theories of all ordinals (see proposition 1.5 below). Finally, let MT[α] be the monadic second order theory of the ordinal α.