Dependent dreams: recounting types
Saharon Shelah

TL;DR
This paper explores the structure and counting of types in dependent theories, extending concepts from stable theories, and establishes bounds on types and indiscernibility properties.
Contribution
It generalizes type decomposition and indiscernibility results from stable to dependent theories, providing bounds on types and ultrafilters.
Findings
Number of types over models is <= lambda in dependent theories
Existence of indiscernible sequences under certain conditions
Bounded number of ultrafilters extending a type
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.…
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Taxonomy
TopicsAdvanced Topology and Set Theory · Limits and Structures in Graph Theory · Computability, Logic, AI Algorithms
