Dependent dreams: recounting types

by Shelah. [Sh:950]

We investigate the class of models of a general dependent theory. We continue [Sh:900] 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 = text {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 bold 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, bar y), the number is finite, with an absolute found (well depending on T and phi). Also if M is saturated then p is the average of an indiscernible sequence inside the model. Lastly, the so-called generic pair conjecture was proved in [Sh:900] for kappa measurable, here it is essentially proved, i.e. for kappa > |T| + beth_omega .

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