Can a small forcing create Kurepa trees

Annals of Pure and Applied Logic 85 (1):47-68 (1997)
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Abstract

In this paper we probe the possibilities of creating a Kurepa tree in a generic extension of a ground model of CH plus no Kurepa trees by an ω1-preserving forcing notion of size at most ω1. In Section 1 we show that in the Lévy model obtained by collapsing all cardinals between ω1 and a strongly inaccessible cardinal by forcing with a countable support Lévy collapsing order, many ω1-preserving forcing notions of size at most ω1 including all ω-proper forcing notions and some proper but not ω-proper forcing notions of size at most ω1 do not create Kurepa trees. In Section 2 we construct a model of CH plus no Kurepa trees, in which there is an ω-distributive Aronszajn tree such that forcing with that Aronszajn tree does create a Kurepa tree in the generic extension. At the end of the paper we ask three questions

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Citations of this work

Club degrees of rigidity and almost Kurepa trees.Gunter Fuchs - 2013 - Archive for Mathematical Logic 52 (1-2):47-66.

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References found in this work

Trees.Thomas J. Jech - 1971 - Journal of Symbolic Logic 36 (1):1-14.
ℵ1-trees.Keith J. Devlin - 1978 - Annals of Mathematical Logic 13 (3):267-330.
Proper Forcing.Saharon Shelah - 1985 - Journal of Symbolic Logic 50 (1):237-239.
Iterated Cohen Extensions and Souslin's Problem.R. M. Solovay & S. Tennenbaum - 1974 - Journal of Symbolic Logic 39 (2):329-330.
Can You Take Solovay's Inaccessible Away?Saharon Shelah & Jean Raisonnier - 1989 - Journal of Symbolic Logic 54 (2):633-635.

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