We extend the theory of “Fine structure and iteration trees” to models having more than one Woodin cardinal.

It is proved that in the absence of proper class inner models with Woodin cardinals, for each n ε {1,…,ω}, ∑3 + n1 absoluteness implies there are n strong cardinals in K (where this denotes a suitably defined global version of the core model for one Woodin cardinal as exposed by Steel. Combined with a forcing argument of Woodin, this establishes that the consistency strength of ∑3 + n1 absoluteness is exactly that of n strong cardinals so that in particular (...) projective absoluteness is equiconsistent with the existence of infinitely many strong cardinals. It is also argued how this theorem is to be construed as the first step in the long range program of showing that projective determinacy is equivalent to its analytical consequences for the projective sets which would settle positively a conjecture of Woodin and thereby solve the last Delfino problem. (shrink)

We introduce two new iteration games: the game , which is a strengthening of the weak iteration game, and the game , which is somewhat stronger than but weaker than the full iteration game of length ω1. For a countable M elementarily embeddable in some Vη, with two players I and II, we can show that II wins and that I does not win.

In [4], Kunen used iterated ultrapowers to show that ifUis a normalκ-complete nontrivial ultrafilter on a cardinalκthenL[U], the class of sets constructive fromU, has only the ultrafilterU∩L[U] and this ultrafilter depends only onκ. In this paper we extend Kunen's methods to arbitrary sequencesUof ultrafilters and obtain generalizations of these results. In particular we answer Problem 1 of Kunen and Paris [5] which asks whether the number of ultrafilters onκcan be intermediate between 1 and 22κ. If there is a normalκ-complete ultrafilterUonκsuch (...) that {α <κ: α is measurable} ∈Uthen there is an inner model with exactly two normal ultrafilters onκ, and ifκis super-compact then there are inner models havingκ+ +,κ+or any cardinal less than or equal toκnormal ultrafilters.These methods also show that several properties ofLwhich had been shown to hold forL[U] also hold forL[U]: using an idea of Silver we show that inL[U] the generalized continuum hypothesis is true, there is a Souslin tree, and there is awell-ordering of the reals. In addition we generalize a result of Kunen to characterize the countaby complete ultrafilters ofL[U]. (shrink)