Can a small forcing create Kurepa trees

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 ω 1-preserving forcing notion of size at most ω 1. In Section 1 we show that in the Lévy model obtained by collapsing all cardinals between ω 1 and a strongly inaccessible cardinal by forcing with a countable support Lévy collapsing order, many ω 1-preserving forcing notions of size at most ω 1 including all ω-proper forcing notions and some proper but not ω-proper forcing notions of size at most ω 1 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.