Abstract. We present an approach to forcing with finite sequences of models that uses models of two types. This approach builds on earlier work of Friedman and Mitchell on forcing to add clubs in cardinals larger than ℵ 1 , with finite conditions. We use the two-type approach to give a new proof of the consistency of the proper forcing axiom. The new proof uses a finite support forcing, as opposed to the countable support iteration in the standard proof. The distinction is important since a proof using finite supports is more amenable to generalizations to cardinals greater than ℵ 1 . MSC-2010: 03E35.

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In this paper I shall present a method for proving determinacy from large cardinals which, in many cases, seems to yield optimal results. One of the main applications extends theorems of Martin, Steel and Woodin about determinacy within the projective hierarchy. The method can also be used to give a new proof of Woodin's theorem about determinacy in L(ℝ).The reason we look for optimal determinacy proofs is not only vanity. Such proofs serve to tighten the connection between large cardinals and descriptive set theory, letting us bring our knowledge of one subject to bear on the other, and thus increasing our understanding of both. A classic example of this is the Harrington-Martin proof that -determinacy implies -determinacy. This is an example of a transfer theorem, which assumes a certain determinacy hypothesis and proves a stronger one. While the statement of the theorem makes no mention of large cardinals, its proof goes through 0#, first proving that-determinacy ⇒ 0# exists,and then that0# exists ⇒ -determinacyMore recent examples of the connection between large cardinals and descriptive set theory include Steel's proof thatADL(ℝ) ⇒ HODL(ℝ) ⊨ GCH,see [9], and several results of Woodin about models of AD+, a strengthening of the axiom of determinacy AD which Woodin has introduced. These proofs not only use large cardinals, but also reveal a deep, structural connection between descriptive set theoretic notions and notions related to large cardinals.

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Abstract. The tree property at κ + states that there are no Aronszajn trees on κ + , or, equivalently, that every κ + tree has a cofinal branch. For singular strong limit cardinals κ, there is tension between the tree property at κ + and failure of the singular cardinal hypothesis at κ; the former is typically the result of the presence of strongly compact cardinals in the background, and the latter is impossible above strongly compacts. In this paper we reconcile the two. We prove from large cardinals that the tree property at κ + is consistent with failure of the singular cardinal hypothesis at κ. §1. Introduction. In the early 1980s Woodin asked whether failure of the singular cardinal hypothesis (SCH) at ℵ ω implies the existence of an Aronszajn tree on ℵ ω+1 . More generally, in 1989 Woodin and others asked whether failure of the SCH at a cardinal κ of cofinality ω, implies the existence of an Aronszajn tree on κ + , see Foreman [7, §2]. To understand the motivation for the question let us recall some results surrounding the SCH and trees in infinitary combinatorics.The singular cardinal hypothesis, in its most specific form, states that 2 κ = κ + whenever κ is a singular strong limit cardinal. (There are several forms that are more general. For example the statement that κ cof(κ) = κ + whenever κ is singular and 2 cof(κ) < κ. Or the statement that for every singular cardinal κ, 2 κ is as small as it can be, subject to two requirements: monotonicity, namely that 2 κ ≥ sup{2 δ | δ < κ}; and König's theorem, which implies cof(2 κ ) > κ. Both these forms imply the specific form, that 2 κ = κ + whenever κ is a singular strong limit cardinal.)Cohen forcing of course shows that the parallel hypothesis for regular cardinals is consistently false. Indeed it can be made to fail in any arbitrary way, subject to monotonicity and König's theorem. For a while after the introduction of forcing it was expected that the same should hold for the SCH, and that proving this was only a matter of discovering sufficiently sophisticated forcing notions. Some progress was made in this direction, and ultimately led to models with failure of the SCH described below. But it turned out that changing the power of a singular cardinal is much harder than changing the power of a regular cardinal, and in some cases it is outright impossible. The first indication of this was a theorem of Silver [25], that the continuum hypothesis cannot fail for the first time at a singular cardinal of uncountable cofinality. Another is a theorem of Solovay [27], that the SCH holds above a strongly compact. The most celebrated is a theorem of Shelah [23], that if 2 ℵn < ℵ ω for each n < ω, then 2 ℵω < ℵ ω4 . Still, the SCH can be made to fail. One way to violate the hypothesis is to start with a measurable cardinal κ, make the continuum hypothesis fail at κ, for example by increasing the power set of κ to κ ++ -an easy task as κ is regular, not singular-and then, assuming κ remained measurable, use Prikry forcing to turn its cofinality to ω. In the resulti...

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Assuming ω supercompact cardinals we force to obtain a model where the tree property holds both at אω+1, and at אn for all 2 ≤ n < ω. A model with the former was obtained by Magidor–Shelah from a large cardinal assumption above a huge cardinal, and recently by Sinapova from ω supercompact cardinals. A model with the latter was obtained by Cummings–Foreman from ω supercompact cardinals. Our model, where the two hold simultaneously, is another step toward the goal of obtaining the tree property on increasingly large intervals of successor cardinals.

We show that every sufficiently iterable countable mouse has a unique iteration strategy whose associated iteration maps are lexicographically minimal. This enables us to extend the results of [3] on the good behavior of the standard parameter from tame mice to arbitrary mice.