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The differences between Kurepa trees and Jech-Kunen trees

The differences between Kurepa trees and Jech-Kunen trees By an ω1 we mean a tree of power ω1 and height ω1. An ω1-tree is called a Kurepa tree if all its levels are countable and it has more than ω1 branches. An ω1-tree is called a Jech-Kunen tree if it has κ branches for some κ strictly between ω1 and $$2^{\omega _1 }$$ . In Sect. 1, we construct a model ofCH plus $$2^{\omega _1 } > \omega _2$$ , in which there exists a Kurepa tree with not Jech-Kunen subtrees and there exists a Jech-Kunen tree with no Kurepa subtrees. This improves two results in [Ji1] by not only eliminating the large cardinal assumption for [Ji1, Theorem 2] but also handling two consistency proofs of [Ji1, Theorem 2 and Theorem 3] simultaneously. In Sect. 2, we first prove a lemma saying that anAxiom A focing of size ω1 over Silver's model will not produce a Kurepa tree in the extension, and then we apply this lemma to prove that, in the model constructed for Theorem 2 in [Ji1], there exists a Jech-Kunen tree and there are no Kurepa trees. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Mathematical Logic Springer Journals

The differences between Kurepa trees and Jech-Kunen trees

Jin, Renling
Archive for Mathematical Logic , Volume 32 (5) – Mar 17, 2005
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References (9)

Publisher
Springer Journals
Copyright
Copyright © 1993 by Springer-Verlag
Subject
Mathematics; Mathematical Logic and Foundations; Mathematics, general; Algebra
ISSN
0933-5846
eISSN
1432-0665
DOI
10.1007/BF01409969
Publisher site
See Article on Publisher Site

Abstract

By an ω1 we mean a tree of power ω1 and height ω1. An ω1-tree is called a Kurepa tree if all its levels are countable and it has more than ω1 branches. An ω1-tree is called a Jech-Kunen tree if it has κ branches for some κ strictly between ω1 and $$2^{\omega _1 }$$ . In Sect. 1, we construct a model ofCH plus $$2^{\omega _1 } > \omega _2$$ , in which there exists a Kurepa tree with not Jech-Kunen subtrees and there exists a Jech-Kunen tree with no Kurepa subtrees. This improves two results in [Ji1] by not only eliminating the large cardinal assumption for [Ji1, Theorem 2] but also handling two consistency proofs of [Ji1, Theorem 2 and Theorem 3] simultaneously. In Sect. 2, we first prove a lemma saying that anAxiom A focing of size ω1 over Silver's model will not produce a Kurepa tree in the extension, and then we apply this lemma to prove that, in the model constructed for Theorem 2 in [Ji1], there exists a Jech-Kunen tree and there are no Kurepa trees.

Journal

Archive for Mathematical Logic – Springer Journals

Published: Mar 17, 2005

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