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

Saharon Shelah; R. Jin

Fundamenta Mathematicae (1992)

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

Abstract

top
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].

How to cite

top

Shelah, 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
  1. [Je1] T. Jech, Trees, J. Symbolic Logic 36 (1971), 1-14. 
  2. [Je2] T. Jech, Set Theory, Academic Press, New York 1978. 
  3. [Je3] T. Jech, Multiple Forcing, Cambridge University Press, 1986. Zbl0601.03019
  4. [Ji1] R. Jin, Some independence results related to the Kurepa tree, Notre Dame J. Formal Logic 32 (1991), 448-457. Zbl0748.03034
  5. [Ji2] R. Jin, A model in which every Kurepa tree is thick, ibid. 33 (1992), 120-125. Zbl0790.03048
  6. [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. 
  7. [K1] K. Kunen, On the cardinality of compact spaces, Notices Amer. Math. Soc. 22 (1975), 212. 
  8. [K2] K. Kunen, Set Theory. An Introduction to Independence Proofs, North-Holland, Amsterdam 1980. 
  9. [S1] S. Shelah, Proper Forcing, Springer, 1982. 
  10. [S2] S. Shelah, new version of Proper Forcing, to appear. 
  11. [SJ] S. Shelah and R. Jin, A model in which there are Jech-Kunen trees but there are no Kurepa trees, preprint. Zbl0790.03049
  12. [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. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.