Strongly dependent theories

By Saharon Shelah. 2005

143 citations recorded by the source corpus.

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 to superstability among stable theory, though are different from it even for stable theories. We show equivalence of some of their definitions, investigate relevant ranks and give some examples, e.g., the first order theory of the p-adics is strongly dependent. The most notable result is: if |A| + |T| ≤ µ, I ⊆ ℭ and |I|≥ℶ|T|+(µ), then some J ⊆ I of cardinality µ+ is an indiscernible sequence over A.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a0bf2e0f1433d54db57c283a8c826559f9fcb9d3

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Strong theories, burden, and weight (2007, background): “The following result shows that strongness is in fact a very strong and natural condition for NIP theories [ 13 ].”
Introduction to theories without the independence property (2008, background): “Using certain ranks which are defined in strongly dependent theories, Shelah achieved the very fundamental result that every sufficiently long sequence in such a theory has a subsequence which is indiscernible [49].”
Strong theories of ordered abelian groups (2015, background): “ry to be “strongly dependent,” with corresponding ordinal-valued ranks, in an attempt to generalize the useful superstable / strictly stable dichotomy to the class of dependent (or NIP) theories, see [23]. However, it should be emphasized that “strongly dependent” really generalizes the idea that ty”
Additivity of the dp-rank (2011, methodology): “ll p(x) or, equivalently, no randomness pattern of depth ω) only needs to be veriﬁed in one variable. Summarizing, we get the following (very) easy corollary, which was originally proved by Shelah in [12] (Observation 1.6). Corollary 4.12. A theory Tis strongly dependent if and only if all the one v”
Externally definable sets and dependent pairs II (2012, background): “e. ∗Supported by the Marie Curie Initial Training Network in Mathematical Logic - MALOA - From MAthematical LOgic to Applications, PITN-GA-2009-238381 1 Assume now that T is NIP. A theorem of Shelah ([She05]), generalizing a result of Poizat and Baizalov in the o-minimal case ([BP98]), states that t”