Notions around tree property 1

Byunghan Kim; Hyeung Joon Kim

doi:10.1016/j.apal.2011.02.001

Notions around tree property 1

Byunghan Kim, Hyeung Joon Kim

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

17 Citations (Scopus)

Abstract

In this paper, we study the notions related to tree property 1 (=TP₁), or, equivalently, SOP₂. Among others, we supply a type-counting criterion for TP₁ and show the equivalence of TP₁ and k- TP₁. Then we introduce the notions of weak k- TP₁ for k≥2, and also supply type-counting criteria for those. We do not know whether weak k- TP₁ implies TP₁, but at least we prove that each weak k- TP₁ implies SOP₁. Our generalization of the tree-indiscernibility results in Džamonja and Shelah (2004)[5] is crucially used throughout the paper.

Original language	English
Pages (from-to)	698-709
Number of pages	12
Journal	Annals of Pure and Applied Logic
Volume	162
Issue number	9
DOIs	https://doi.org/10.1016/j.apal.2011.02.001
Publication status	Published - 2011 Sept

Bibliographical note

Funding Information:
The first author was supported by NRF grant 2010-0016044. The second author was supported by Hi Seoul Science/Humanities Fellowship from Seoul Scholarship Foundation, and by the Brain Korea 21 Program 2006–2009. This paper is a part of the second author’s Ph.D. thesis.

All Science Journal Classification (ASJC) codes

Logic

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10.1016/j.apal.2011.02.001

Cite this

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AB - In this paper, we study the notions related to tree property 1 (=TP1), or, equivalently, SOP2. Among others, we supply a type-counting criterion for TP1 and show the equivalence of TP1 and k- TP1. Then we introduce the notions of weak k- TP1 for k≥2, and also supply type-counting criteria for those. We do not know whether weak k- TP1 implies TP1, but at least we prove that each weak k- TP1 implies SOP1. Our generalization of the tree-indiscernibility results in Džamonja and Shelah (2004)[5] is crucially used throughout the paper.

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Notions around tree property 1

Abstract

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