Essential Kurepa trees versus essential Jech-Kunen trees

Renling Jin; Saharon Shelah

doi:10.1016/0168-0072(94)90021-3

Essential Kurepa trees versus essential Jech-Kunen trees

Renling Jin^*, Saharon Shelah

^*Corresponding author for this work

Einstein Institute of Mathematics

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

2 Scopus citations

Abstract

By an ω₁-tree we mean a tree of cardinality 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^ω₁. A Kurepa tree is called an essential Kurepa tree if it contains no Jech-Kunen subtrees. A Jech-Kunen tree is called an essential Jech-Kunen tree if it is no Kurepa subtrees. In this paper we prove that (1) it is consistent with CH and 2^ω₁ #62; ω₂ that there exist essential Kurepa trees and there are no essential Jech-Kunen trees, (2) it is consistent with CH and 2^ω₁ #62; ω₂ plus the existence of a Kurepa tree with 2^ω₁ branches that there exist essential Jech-Kunen trees and there are no essential Kurepa trees. In the second result we require the existence of a Kurepa tree with 2^ω₁ branches in order to avoid triviality.

Original language	English
Pages (from-to)	107-131
Number of pages	25
Journal	Annals of Pure and Applied Logic
Volume	69
Issue number	1
DOIs	https://doi.org/10.1016/0168-0072(94)90021-3
State	Published - 6 Sep 1994

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title = "Essential Kurepa trees versus essential Jech-Kunen trees",

abstract = "By an ω1-tree we mean a tree of cardinality 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 2ω1. A Kurepa tree is called an essential Kurepa tree if it contains no Jech-Kunen subtrees. A Jech-Kunen tree is called an essential Jech-Kunen tree if it is no Kurepa subtrees. In this paper we prove that (1) it is consistent with CH and 2ω1 \#62; ω2 that there exist essential Kurepa trees and there are no essential Jech-Kunen trees, (2) it is consistent with CH and 2ω1 \#62; ω2 plus the existence of a Kurepa tree with 2ω1 branches that there exist essential Jech-Kunen trees and there are no essential Kurepa trees. In the second result we require the existence of a Kurepa tree with 2ω1 branches in order to avoid triviality.",

author = "Renling Jin and Saharon Shelah",

year = "1994",

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doi = "10.1016/0168-0072(94)90021-3",

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T1 - Essential Kurepa trees versus essential Jech-Kunen trees

AU - Jin, Renling

AU - Shelah, Saharon

PY - 1994/9/6

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N2 - By an ω1-tree we mean a tree of cardinality 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 2ω1. A Kurepa tree is called an essential Kurepa tree if it contains no Jech-Kunen subtrees. A Jech-Kunen tree is called an essential Jech-Kunen tree if it is no Kurepa subtrees. In this paper we prove that (1) it is consistent with CH and 2ω1 #62; ω2 that there exist essential Kurepa trees and there are no essential Jech-Kunen trees, (2) it is consistent with CH and 2ω1 #62; ω2 plus the existence of a Kurepa tree with 2ω1 branches that there exist essential Jech-Kunen trees and there are no essential Kurepa trees. In the second result we require the existence of a Kurepa tree with 2ω1 branches in order to avoid triviality.

AB - By an ω1-tree we mean a tree of cardinality 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 2ω1. A Kurepa tree is called an essential Kurepa tree if it contains no Jech-Kunen subtrees. A Jech-Kunen tree is called an essential Jech-Kunen tree if it is no Kurepa subtrees. In this paper we prove that (1) it is consistent with CH and 2ω1 #62; ω2 that there exist essential Kurepa trees and there are no essential Jech-Kunen trees, (2) it is consistent with CH and 2ω1 #62; ω2 plus the existence of a Kurepa tree with 2ω1 branches that there exist essential Jech-Kunen trees and there are no essential Kurepa trees. In the second result we require the existence of a Kurepa tree with 2ω1 branches in order to avoid triviality.

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Essential Kurepa trees versus essential Jech-Kunen trees

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