83 research outputs found
Theories without the tree property of the second kind
We initiate a systematic study of the class of theories without the tree
property of the second kind - NTP2. Most importantly, we show: the burden is
"sub-multiplicative" in arbitrary theories (in particular, if a theory has TP2
then there is a formula with a single variable witnessing this); NTP2 is
equivalent to the generalized Kim's lemma and to the boundedness of ist-weight;
the dp-rank of a type in an arbitrary theory is witnessed by mutually
indiscernible sequences of realizations of the type, after adding some
parameters - so the dp-rank of a 1-type in any theory is always witnessed by
sequences of singletons; in NTP2 theories, simple types are co-simple,
characterized by the co-independence theorem, and forking between the
realizations of a simple type and arbitrary elements satisfies full symmetry; a
Henselian valued field of characteristic (0,0) is NTP2 (strong, of finite
burden) if and only if the residue field is NTP2 (the residue field and the
value group are strong, of finite burden respectively), so in particular any
ultraproduct of p-adics is NTP2; adding a generic predicate to a geometric NTP2
theory preserves NTP2.Comment: 35 pages; v.3: a discussion and a Conjecture 2.7 on the
sub-additivity of burden had been added; Section 3.1 on the SOPn hierarchy
restricted to NTP2 theories had been added; Problem 7.13 had been updated;
numbering of theorems had been changed and some minor typos were fixed;
Annals of Pure and Applied Logic, accepte
Externally definable sets and dependent pairs
We prove that externally definable sets in first order NIP theories have
honest definitions, giving a new proof of Shelah's expansion theorem. Also we
discuss a weak notion of stable embeddedness true in this context. Those
results are then used to prove a general theorem on dependent pairs, which in
particular answers a question of Baldwin and Benedikt on naming an
indiscernible sequence.Comment: 17 pages, some typos and mistakes corrected, overall presentation
improved, more details for the examples are give
Henselian valued fields and inp-minimality
We prove that every ultraproduct of -adics is inp-minimal (i.e., of burden
). More generally, we prove an Ax-Kochen type result on preservation of
inp-minimality for Henselian valued fields of equicharacteristic in the RV
language.Comment: v.2: 15 pages, minor corrections and presentation improvements;
accepted to the Journal of Symbolic Logi
Naming an indiscernible sequence in NIP theories
In this short note we show that if we add predicate for a dense complete
indiscernible sequence in a dependent theory then the result is still
dependent. This answers a question of Baldwin and Benedikt and implies that
every unstable dependent theory has a dependent expansion interpreting linear
order.Comment: 3 page
An independence theorem for NTP2 theories
We establish several results regarding dividing and forking in NTP2 theories.
We show that dividing is the same as array-dividing. Combining it with
existence of strictly invariant sequences we deduce that forking satisfies the
chain condition over extension bases (namely, the forking ideal is S1, in
Hrushovski's terminology). Using it we prove an independence theorem over
extension bases (which, in the case of simple theories, specializes to the
ordinary independence theorem). As an application we show that Lascar strong
type and compact strong type coincide over extension bases in an NTP2 theory.
We also define the dividing order of a theory -- a generalization of Poizat's
fundamental order from stable theories -- and give some equivalent
characterizations under the assumption of NTP2. The last section is devoted to
a refinement of the class of strong theories and its place in the
classification hierarchy
External definability and groups in NIP theories
We prove that many properties and invariants of definable groups in NIP
theories, such as definable amenability, G/G^{00}, etc., are preserved when
passing to the theory of the Shelah expansion by externally definable sets,
M^{ext}, of a model M. In the light of these results we continue the study of
the "definable topological dynamics" of groups in NIP theories. In particular
we prove the Ellis group conjecture relating the Ellis group to G/G^{00} in
some new cases, including definably amenable groups in o-minimal structures.Comment: 28 pages. Introduction was expanded and some minor mistakes were
corrected. Journal of the London Mathematical Society, accepte
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