Topology in Zfc*

The first-order theory of linear order is far from trivia!. The monadic (secondorder) theory of linear order is much stronger. Stil~ surpr,Mng decidability results were proven in that direction. Rabin proved in [10i that the monadic theory of the rational chain is decidable. Hence the monadic theory of all countable linear orders is decidable. Biichi proved in [I] that the monadic theory of ordinals of cardinality at most Nt is decidable. The decision problems for the monadic theory of the real line R, the monadic theory of linear order, the monadic theory of Nz and the monadic theory of ordinals were long open. The last two theories are taken care of in [5] and will not be discussed here. Let us recall the definition of the monadic theory of order. The pure monadic (second-order) language has two sorts of variables: for points and fo:: sets of points, Its atomic formulas have the form x~ ~ X/. The :est of its tormulas are built from the atomic ones by means of ordinary propositional connectives and quantifiers for variables of either sort. Augmenting the pure monadic language by the symbol < for an order on points we get the monadic lang,:age of o~"der; the new atomic formulas have the form :q <X~. For the sake of brevity linearly ordered chains are here called chains. The monadic theory of a chain C is the theory of C in the monadic language of order when the set variables range over all subsets of C. One specific chain of special interest to us is the real line/i '. Recall that a set of reals is called meager if it is a union of ~<No nowhere dense sets. We call it pseudo-meager if it is a union of <2 ~o nowhere dense sets. The Contimmm