I will show that Diamond Plus holds in inner models of the form L[A], for subsets A of aleph one in the sense of L[A]. Putting this together with the result from last meeting, that Diamond Plus implies the Kurepa Hypothesis, I will show that if the Kurepa Hypothesis fails, then aleph two is an inaccessible cardinal in L. Again, putting this together with another result from the previous seminar meeting, that one can force the failure of Kurepa’s Hypothesis over a model with an inaccessible cardinal, this shows the equiconsistency of the failure of Kurepa’s Hypothesis with an inaccessible cardinal, over ZFC. These results are mainly due to Silver and Solovay.

The speaker will prove some connections between Diamond Plus and the Kurepa Hypothesis.