A Model in Which There Are Jech–kunen Trees but There Are No Kurepa Trees

By an ω1–tree we mean a tree of power ω1 and height ω1. We call an ω1–tree a Jech–Kunen tree if it has κ–many branches for some κ strictly between ω1 and 21 . In this paper we construct the models of CH plus 21 > ω2, in which there are Jech–Kunen trees and there are no Kurepa trees. An partially ordered set, or poset for short, 〈T, ω1 and for every α ∈ ω1, |Tα| < ω1. An ω1–tree is called a Jech–Kunen tree if ω1 < |B(T )| < 2 1. The independence of the existence of Kurepa trees was proved by J. H. Silver (see [K2, §3 of Chapter VIII]). T. Jech in [Je1] constructed by forcing a model of CH plus 21 > ω2, in which there is a Jech–Kunen tree. In fact, it is a Kurepa tree with fewer than 21–many branches. The independence of the existence of Jech–Kunen trees under CH plus 21 > ω2 was given by K. Kunen [K1]. In his paper he gave an equivalent form of Jech–Kunen trees in terms of compact Hausdorff spaces. The detailed proof can be found in [Ju, Theorem 4.8]. In both Silver and Kunen’s proofs, the existence of a strongly inaccessible cardinal was assumed (the assumption is also necessary). The technique they used to kill all Kurepa trees or Jech–Kunen trees is to show that if an ω1–tree T has a new branch in an ω1–closed forcing extension, then T must have a subtree which is isomorphic to 〈21,⊆〉, a complete binary tree of height ω1. So in Kunen’s model not only all Jech–Kunen trees are killed, but also all Kurepa trees are killed. R. Jin in [Ji1] started discussing the differences between Kurepa trees and Jech– Kunen trees. He showed that it is independent of CH plus 21 > ω2 that there 1980 Mathematics Subject Classification (1985 Revision). Primary 03E35. The research of the first author was partially supported by the Basic Research Fund, Israeli Acad. of Science Publ. nu. 466.