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

Shelah, Saharon; Jin, R.

Shelah, Saharon ; Jin, R.

Fundamenta Mathematicae, Tome 141 (1992), p. 287-296 / Harvested from The Polish Digital Mathematics Library

Résumé

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].

Publié le : 1992-01-01

Zbl 0809.03038

EUDML-ID : urn:eudml:doc:211967

@article{bwmeta1.element.bwnjournal-article-fmv141i3p287bwm,
     author = {Saharon Shelah and R. Jin},
     title = {Planting Kurepa trees and killing Jech-Kunen trees in a model by using one inaccessible cardinal},
     journal = {Fundamenta Mathematicae},
     volume = {141},
     year = {1992},
     pages = {287-296},
     zbl = {0809.03038},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv141i3p287bwm}
}

Shelah, Saharon; Jin, R. Planting Kurepa trees and killing Jech-Кunen trees in a model by using one inaccessible cardinal. Fundamenta Mathematicae, Tome 141 (1992) pp. 287-296. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv141i3p287bwm/

Bibliographie

[00000] [Je1] T. Jech, Trees, J. Symbolic Logic 36 (1971), 1-14.

[00001] [Je2] T. Jech, Set Theory, Academic Press, New York 1978.

[00002] [Je3] T. Jech, Multiple Forcing, Cambridge University Press, 1986. | Zbl 0601.03019

[00003] [Ji1] R. Jin, Some independence results related to the Kurepa tree, Notre Dame J. Formal Logic 32 (1991), 448-457. | Zbl 0748.03034

[00004] [Ji2] R. Jin, A model in which every Kurepa tree is thick, ibid. 33 (1992), 120-125. | Zbl 0790.03048

[00005] [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.

[00006] [K1] K. Kunen, On the cardinality of compact spaces, Notices Amer. Math. Soc. 22 (1975), 212.

[00007] [K2] K. Kunen, Set Theory. An Introduction to Independence Proofs, North-Holland, Amsterdam 1980.

[00008] [S1] S. Shelah, Proper Forcing, Springer, 1982.

[00009] [S2] S. Shelah, new version of Proper Forcing, to appear.

[00010] [SJ] S. Shelah and R. Jin, A model in which there are Jech-Kunen trees but there are no Kurepa trees, preprint. | Zbl 0790.03049

[00011] [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.