We present a general construction of a Îº-sequence in Jensenâ€™s fine structural extender models. This construction yields a local definition of a canonical Îº-sequence as well as a characterization ofâ€¦ (More)

We say that a cardinal Î´ is countably closed if Î³ â„µ 0 < Î´ whenever Î³ < Î´. In this paper, we prove the following theorem. Theorem 1.1. Suppose that â„¦ is a measurable cardinal, and that there is noâ€¦ (More)

We show that either of the following hypotheses imply that there is an inner model with a proper class of strong cardinals and a proper class of Woodin cardinals. 1) There is a countably closedâ€¦ (More)

The main result of [MiSchSt] is that Theorem 0.1 holds under the additional assumption that card(Îº) is countably closed. But often, in applications, countable closure is not available. Theorem 0.1â€¦ (More)

We introduce the weak partial square principles pÎ»,Îº and p Î»,<Îº, which combine the ideas of a weak square sequence and a partial square sequence. We construct models in which weak partial squareâ€¦ (More)

We define what it means for a function on Ï‰1 to be a collapsing function for Î» and show that if there exists a collapsing function for (21), then there is no precipitous ideal on Ï‰1. We show that aâ€¦ (More)

We prove new upper bound theorems on the consistency strengths of SPFA(Î¸), SPFA(Î¸-linked) and SPFA(Î¸-cc). Our results are in terms of (Î¸,Î“)-subcompactness, which is a new large cardinal notion thatâ€¦ (More)