Can a small forcing create Kurepa trees?

by Jin and Shelah. [JiSh:563]
Annals Pure and Applied Logic, 1997
In the paper we probe the possibilities of creating a Kurepa tree in a generic extension of a model of CH plus no Kurepa trees by an omega_1-preserving forcing notion of size at most omega_1 . In the first section we show that in the Levy model obtained by collapsing all cardinals between omega_1 and a strongly inaccessible cardinal by forcing with a countable support Levy collapsing order many omega_1-preserving forcing notions of size at most omega_1 including all omega-proper forcing notions and some proper but not omega-proper forcing notions of size at most omega_1 do not create Kurepa trees. In the second section we construct a model of CH plus no Kurepa trees, in which there is an omega-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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