Dependent dreams: recounting types

Abstract

We investigate the class of models of a general dependent theory. We continue math.LO/0702292 in particular investigating so called "decomposition of types"; thesis is that what holds for stable theory and for Th(Q,<) hold for dependent theories. Another way to say this is: we have to look at small enough neighborhood and use reasonably definable types to analyze a type. We note the results understable without reading. First, a parallel to the "stability spectrum", the "recounting of types", that is assume lambda = lambda^{< lambda} is large enough, M a saturated model of T of cardinality lambda, let bold S_{aut}(M) be the number of complete types over M up to being conjugate, i.e. we identify p,q when some automorphism of M maps p to q . Whereas for independent T the number is 2^lambda, for dependent T the number is <= lambda moreover it is <= | alpha |^{|T|} when lambda = aleph_alpha. Second, for stable theories "lots of indiscernibility exists" a "too good indiscernible existence theorem" saying, e.g. that if the type tp (d_beta ; {d_beta : beta < alpha}) is increasing for alpha < kappa = cf(kappa) and kappa > 2^{|T|} then <d_alpha : alpha in S> is indiscernible for some stationary S subseteq kappa. Third, for stable T,a model is kappa-saturated iff it is aleph_epsilon-saturated and every infinite indiscernible set (of elements) of cardinality < kappa can be increased. We prove here an analog. Fourth, for p in S(M), the number of ultrafilters on the outside definable subsets of M extending p has an absolute bound 2^{|T|} . Restricting ourselves to one phi(x, y), the number is finite, with an absolute found (well depending on T and phi).