We show that the reals in the minimal iterable inner model having n Woodin cardinals are precisely those which are A,‘+ 2 definable from some countable ordinal. (One direction here is due to Hugh… (More)

We shall present some results on the the properties of derived models of mice, and on the existence of mice with large derived models. The basic plan of the paper is: Sections 1–5. Introduction.… (More)

We shall exposit here one of the basic theorems leading from large cardinals to determinacy, a result of Woodin known as the derived model theorem. The theorem dates from the mid80’s and has been… (More)

Part of the ambiguity lies in the various points of view from which this question might be considered. The crudest di erence lies between the point of view of the working mathematician and that of… (More)

We say that a cardinal δ is countably closed if γ ℵ 0 < δ whenever γ < δ. In this paper, we prove the following theorem. Theorem 1.1. Suppose that Ω is a measurable cardinal, and that there is no… (More)