EXAMPLES IN DEPENDENT THEORIES

@article{Kaplan2014EXAMPLESID,
  title={EXAMPLES IN DEPENDENT THEORIES},
  author={Itay Kaplan and Saharon Shelah},
  journal={The Journal of Symbolic Logic},
  year={2014},
  volume={79},
  pages={585 - 619}
}
Abstract In the first part we show a counterexample to a conjecture by Shelah regarding the existence of indiscernible sequences in dependent theories (up to the first inaccessible cardinal). In the second part we discuss generic pairs, and give an example where the pair is not dependent. Then we define the notion of directionality which deals with counting the number of coheirs of a type and we give examples of the different possibilities. Then we discuss nonsplintering, an interesting notion… 
8 Citations
DP-MINIMALITY: INVARIANT TYPES AND DP-RANK
TLDR
It is proved that an invariant dp-minimal type is either finitely satisfiable or definable, and it is shown that if the structure expands a divisible ordered abelian group, then d p-rank coincides with the dimension coming from the order.
Title On the number of dedekind cuts and two-cardinal models of dependent theories
For an infinite cardinal κ, let dedκ denote the supremum of the number of Dedekind cuts in linear orders of size κ. It is known that κ < ded κ ≤ 2κ for all κ and that dedκ < 2κ is consistent for any
Invariant types in NIP theories
TLDR
A definable version of the (p,q)-theorem in theories of small or medium directionality is proved and some amalgamation results for invariant types are shown.
Exact saturation in pseudo-elementary classes for simple and stable theories
We study PC-exact saturation for stable and simple theories. Among other results, we show that PC-exact saturation characterizes the stability cardinals of size at least continuum of a countable
LIST OF PUBLICATIONS
1. Sh:a Saharon Shelah. Classification theory and the number of nonisomorphic models, volume 92 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam-New
Stable theories and representation over sets
In this paper we give characterizations of the stable and ℵ0‐stable theories, in terms of an external property called representation. In the sense of the representation property, the mentioned
ON THE AUTOMORPHISM GROUP OF THE UNIVERSAL HOMOGENEOUS MEET-TREE
Abstract We show that the countable universal homogeneous meet-tree has a generic automorphism, but it does not have a generic pair of automorphisms.
ON THE NUMBER OF DEDEKIND CUTS AND TWO-CARDINAL MODELS OF DEPENDENT THEORIES
  • A. Chernikov, S. Shelah
  • Mathematics, Computer Science
    Journal of the Institute of Mathematics of Jussieu
  • 2015
TLDR
The Hanf numbers are calculated for the existence of two-cardinal models with arbitrarily large gaps and for theexistence of arbitrarily large models omitting a type in the class of countable dependent first-order theories to show that these bounds are as large as in theclass of all countable theories.

References

SHOWING 1-10 OF 39 REFERENCES
Dependent theories and the generic pair conjecture
We try to understand complete types over a somewhat saturated model of a complete first-order theory which is dependent (previously called NIP), by "decomposition theorems for such types". Our thesis
Dependent dreams: recounting types
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
Strongly dependent theories
We further investigate the class of models of a strongly dependent (first order complete) theory T, continuing [Sh:715], [Sh:783] and related works. Those are properties (= classes) somewhat parallel
On non-forking spectra
Non-forking is one of the most important notions in modern model theory capturing the idea of a generic extension of a type (which is a far-reaching generalization of the concept of a generic point
A dependent theory with few indiscernibles
We give a full solution to the question of existence of indiscernibles in dependent theories by proving the following theorem: For every θ there is a dependent theory T of size θ such that for all κ
Categoricity in power
Introduction. A theory, 1, (formalized in the first order predicate calculus) is categorical in power K if it has exactly one isomorphism type of models of power K. This notion was introduced by Los
Around classification theory of models
Classifying generalized quantifiers.- Classification over a predicate II.- Existence of endo-rigid Boolean algebras.- On the no(M) for M of singular power.- Non standard uniserial module over a
On NIP and invariant measures
We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [13]. Among key results are
Dependent T and existence of limit models
Abstract Does the class of linear orders have (one of the variants of) the so called (λ;κ)-limit model? It is necessarily unique, and naturally assuming some instances of G.C.H. we get some positive
Six classes of theories
  • H. Keisler
  • Mathematics
    Journal of the Australian Mathematical Society
  • 1976
A theory T is said to κ-stable if, given a pair of models U ⊂ B of T with U of power κ, there are only κ types of elements of B over U (types are defined below). This notion was introduced by Morley
...
...