Theories without the tree property of the second kind

Annals of Pure and Applied Logic 165 (2):695-723 (2014)
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Abstract

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 ; NTP2 is equivalent to the generalized Kimʼs 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 is NTP2 if and only if the residue field is NTP2 , so in particular any ultraproduct of p-adics is NTP2; adding a generic predicate to a geometric NTP2 theory preserves NTP2

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Citations of this work

On model-theoretic tree properties.Artem Chernikov & Nicholas Ramsey - 2016 - Journal of Mathematical Logic 16 (2):1650009.
Generic expansion and Skolemization in NSOP 1 theories.Alex Kruckman & Nicholas Ramsey - 2018 - Annals of Pure and Applied Logic 169 (8):755-774.
Independence over arbitrary sets in NSOP1 theories.Jan Dobrowolski, Byunghan Kim & Nicholas Ramsey - 2022 - Annals of Pure and Applied Logic 173 (2):103058.
Pseudo real closed fields, pseudo p-adically closed fields and NTP2.Samaria Montenegro - 2017 - Annals of Pure and Applied Logic 168 (1):191-232.

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References found in this work

Generic structures and simple theories.Z. Chatzidakis & A. Pillay - 1998 - Annals of Pure and Applied Logic 95 (1-3):71-92.
Forking and dividing in NTP₂ theories.Artem Chernikov & Itay Kaplan - 2012 - Journal of Symbolic Logic 77 (1):1-20.
Toward classifying unstable theories.Saharon Shelah - 1996 - Annals of Pure and Applied Logic 80 (3):229-255.
On dp-minimal ordered structures.Pierre Simon - 2011 - Journal of Symbolic Logic 76 (2):448 - 460.
Dp-Minimality: Basic Facts and Examples.Alfred Dolich, John Goodrick & David Lippel - 2011 - Notre Dame Journal of Formal Logic 52 (3):267-288.

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