Sh:950

• Shelah, S. Dependent dreams: recounting types. Preprint. arXiv: 1202.5795
• Abstract:
  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(\mathbb 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 \mathbf S_{\text{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 \le \lambda moreover it is \le |\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 \langle d_\alpha:\alpha \in S\rangle is indiscernible for some stationary S \subseteq \kappa. Third, for stable T,a model is \kappa-saturated iff it is \aleph_\varepsilon-saturated and every infinite indiscernible set (of elements) of cardinality < \kappa can be increased. We prove here an analog. Fourth, for p \in \mathbf 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 \varphi(x,\bar y), the number is finite, with an absolute found (well depending on T and \varphi). 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.
• Version 2014-05-02_12 (120p)

@article{Sh:950,
 author = {Shelah, Saharon},
 title = {{Dependent dreams: recounting types}},
 note = {\href{https://arxiv.org/abs/1202.5795}{arXiv: 1202.5795}},
 arxiv_number = {1202.5795}
}