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

• Autorzy: Saharon Shelah , R. Jin
• Języki publikacji: EN

Abstrakty

EN

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].

Kategorie tematyczne

• 03E35: Consistency and independence results

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1992
• Tom: 141
• Numer: 3
• Strony: 287-296

Daty

wydano

1992

otrzymano

1992-03-03

Twórcy

autor

Saharon Shelah

• Institute of Mathematics, The Hebrew University, Jerusalem, Israel
• Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903 U.S.A.

autor

R. Jin

• Department of Mathematics, University of California, Berkeley, California 94720, U.S.A.

Bibliografia

• [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.
• [Ji1] R. Jin, Some independence results related to the Kurepa tree, Notre Dame J. Formal Logic 32 (1991), 448-457.
• [Ji2] R. Jin, A model in which every Kurepa tree is thick, ibid. 33 (1992), 120-125.
• [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.
• [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.