Modest Theory of Short Chains

We analyse here the monadic theory of the rational order, the monadic theory of the real line with quantification over "small" subsets and models of these theories. We prove that the results are in some sense the best possible. ?0. Introduction. A chain is a linearly ordered set. A chain is short iff it embeds neither Wi nor O1. Shortness is expressible in the monadic language of order. For convenience, we shall use the term "chain" to mean "short chain". Modest chains are defined in ?2. A chain is modest iff it is p-modest for every positive integer p; p-modesty is expressible in the monadic theory of order. Let R be the real line. Set X c R will be called pseudo-meager iff it is a union of less than c (the cardinality of continuum) nowhere dense subsets. It is well known that the hypothesis "R is not pseudo-meager" can be neither proved nor disproved in ZFC. The Continuum Hypothesis (or Martin's Axiom, see [MS]) implies that each pseudo-meager set is meager. By the Baire Theorem, R is not meager. Let Q be the chain of rational numbers. The monadic theory of Q is decidable (see [Ra]) but not categorical. By [Sh], there exist nonseparable chains monadically equivalent to Q (i.e. having the same monadic theory as Q) and if R is not pseudomeager then there are subchains of R of cardinality c monadically equivalent to Q. THEOREM 1 (SEE ?3). A chain is monadically equivalent to Q iff it is modest and has neither jumps nor endpoints. Moreover there exists an algorithm associating a pair (p, s) with each sentence SD in the monadic language of order in such a way that either = 0 and all p-modest chains without jumps and endpoints satisfy q or c = 1 and all these chains do not satisfy q. Each modest chain is embeddable into a modest chain without jumps and end-