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ON WIDE ARONSZAJN TREES IN THE PRESENCE OF MA

Part of: Set theory

Published online by Cambridge University Press:  07 September 2020

MIRNA DŽAMONJA  and
SAHARON SHELAH
MIRNA DŽAMONJA
Affiliation:
INSTITUT D'HISTOIRE ET DE PHILOSOPHIE DES SCIENCES ET DES TECHNIQUES (IHPST) CNRS & UNIVERSITÉ PANTHÉON-SORBONNE 13 RUE DE FOUR, 75006 PARIS, FRANCE INSTITUTE OF MATHEMATICS CZECH ACADEMY OF SCIENCES ITNÁ 25, 115 76 PRAGUE, CZECH REPUBLIC E-mail: M.Dzamonja@uea.ac.uk URL: mirna.dzamonja@univ-paris1.fr
SAHARON SHELAH
Affiliation:
DEPARTMENT OF MATHEMATICS HEBREW UNIVERSITY OF JERUSALEM 91904, GIVAT RAM, ISRAEL E-mail:shelah@math.huji.ac.il URL: http://shelah.logic.at
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Abstract

A wide Aronszajn tree is a tree of size and height $\omega _{1}$ with no uncountable branches. We prove that under $MA(\omega _{1}\!)$ there is no wide Aronszajn tree which is universal under weak embeddings. This solves an open question of Mekler and Väänänen from 1994.

We also prove that under $MA(\omega _{1}\!)$, every wide Aronszajn tree weakly embeds in an Aronszajn tree, which combined with a result of Todorčević from 2007, gives that under $MA(\omega _{1}\!)$ every wide Aronszajn tree embeds into a Lipschitz tree or a coherent tree. We also prove that under $MA(\omega _{1}\!)$ there is no wide Aronszajn tree which weakly embeds all Aronszajn trees, improving the result in the first paragraph as well as a result of Todorčević from 2007 who proved that under $MA(\omega _{1}\!)$ there are no universal Aronszajn trees.

Keywords

wide Aronszajn treeMartin axiomuniversality

MSC classification

Primary: 03E05: Other combinatorial set theory
Secondary: 03E35: Consistency and independence results 03E50: Continuum hypothesis and Martin's axiom

Information

Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 1 , March 2021 , pp. 210 - 223
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Copyright
© The Association for Symbolic Logic 2020

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References

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