Random Trees Under CH

James Hirschorn

We extend Jensen’s Theorem that Suslin’s Hypothesis is consistent with CH, by showing that the statement Suslin’s Hypothesis holds in any forcing extension by a measure algebra is consistent with CH. In doing so we prove that a variation of the principle (∗) (see [AT97] and [Tod99]) for closed sets of ordinals is consistent with GCH. We also extend Silver’s Theorem that, assuming the existence of an inaccessible cardinal, the failure of Kurepa’s Hypothesis is consistent with CH, by showing that the statement Kurepa’s Hypothesis fails in any forcing extension by a measure algebra is consistent with CH.

[AT97] Uri Abraham and Stevo Todorčević, Partition properties of ω_{1} compatible with CH, Fund. Math. 152 (1997), no. 2, 165–181.

[Tod99] Stevo Todorčević, A dichotomy for P -ideals of countable sets, preprint, 1999.

Address: University of Toronto. Postal Address: 409 Huron St. #210, Toronto, ON M5S 2G5, Canada

E-mail: hirschor@math.toronto.edu