# Planting Kurepa trees and killing Jech-Кunen trees in a model by using one inaccessible cardinal

Fundamenta Mathematicae (1992)

- Volume: 141, Issue: 3, page 287-296
- ISSN: 0016-2736

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topShelah, Saharon, and Jin, R.. "Planting Kurepa trees and killing Jech-Кunen trees in a model by using one inaccessible cardinal." Fundamenta Mathematicae 141.3 (1992): 287-296. <http://eudml.org/doc/211967>.

@article{Shelah1992,

abstract = {By an $ω_1$- tree we mean a tree of power $ω_1$ and height $ω_1$. Under CH and $2^\{ω_\{1\}\} > ω_2$ we call an $ω_1$-tree a Jech-Kunen tree if it has κ-many branches for some κ strictly between $ω_1$ and $2^\{ω_\{1\}\}$. In this paper we prove that, assuming the existence of one inaccessible cardinal, (1) it is consistent with CH plus $2^\{ω_\{1\}\} > ω_2$ that there exist Kurepa trees and there are no Jech-Kunen trees, which answers a question of [Ji2], (2) it is consistent with CH plus $2^\{ω_\{1\}\} = ω_4$ that there only exist Kurepa trees with $ω_\{3\}$-many branches, which answers another question of [Ji2].},

author = {Shelah, Saharon, Jin, R.},

journal = {Fundamenta Mathematicae},

keywords = {-tree; inaccessible cardinal; Kurepa trees; Jech-Kunen trees},

language = {eng},

number = {3},

pages = {287-296},

title = {Planting Kurepa trees and killing Jech-Кunen trees in a model by using one inaccessible cardinal},

url = {http://eudml.org/doc/211967},

volume = {141},

year = {1992},

}

TY - JOUR

AU - Shelah, Saharon

AU - Jin, R.

TI - Planting Kurepa trees and killing Jech-Кunen trees in a model by using one inaccessible cardinal

JO - Fundamenta Mathematicae

PY - 1992

VL - 141

IS - 3

SP - 287

EP - 296

AB - By an $ω_1$- tree we mean a tree of power $ω_1$ and height $ω_1$. Under CH and $2^{ω_{1}} > ω_2$ we call an $ω_1$-tree a Jech-Kunen tree if it has κ-many branches for some κ strictly between $ω_1$ and $2^{ω_{1}}$. In this paper we prove that, assuming the existence of one inaccessible cardinal, (1) it is consistent with CH plus $2^{ω_{1}} > ω_2$ that there exist Kurepa trees and there are no Jech-Kunen trees, which answers a question of [Ji2], (2) it is consistent with CH plus $2^{ω_{1}} = ω_4$ that there only exist Kurepa trees with $ω_{3}$-many branches, which answers another question of [Ji2].

LA - eng

KW - -tree; inaccessible cardinal; Kurepa trees; Jech-Kunen trees

UR - http://eudml.org/doc/211967

ER -

## References

top- [Je1] T. Jech, Trees, J. Symbolic Logic 36 (1971), 1-14.
- [Je2] T. Jech, Set Theory, Academic Press, New York 1978.
- [Je3] T. Jech, Multiple Forcing, Cambridge University Press, 1986. Zbl0601.03019
- [Ji1] R. Jin, Some independence results related to the Kurepa tree, Notre Dame J. Formal Logic 32 (1991), 448-457. Zbl0748.03034
- [Ji2] R. Jin, A model in which every Kurepa tree is thick, ibid. 33 (1992), 120-125. Zbl0790.03048
- [Ju] I. Juhász, Cardinal functions II, in: Handbook of Set-Theoretic Topology, K. Kunen and J. E. Vaughan (eds.), North-Holland, Amsterdam 1984, 63-110.
- [K1] K. Kunen, On the cardinality of compact spaces, Notices Amer. Math. Soc. 22 (1975), 212.
- [K2] K. Kunen, Set Theory. An Introduction to Independence Proofs, North-Holland, Amsterdam 1980.
- [S1] S. Shelah, Proper Forcing, Springer, 1982.
- [S2] S. Shelah, new version of Proper Forcing, to appear.
- [SJ] S. Shelah and R. Jin, A model in which there are Jech-Kunen trees but there are no Kurepa trees, preprint. Zbl0790.03049
- [T] S. Todorčević, Trees and linearly ordered sets, in: Handbook of Set-Theoretic Topology, K. Kunen and J. E. Vaughan (eds.), North-Holland, Amsterdam 1984, 235-293.

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