This paper concerns the model of Cummings and Foreman where from ω supercompact cardinals they obtain the tree property at each אn for 2 ≤ n < ω. We prove some structural facts about this model. We show that the combinatorics at אω+1 in this model depend strongly on the properties of ω1 in the ground model. From different ground models for the CummingsForeman iteration we can obtain either אω+1...

We present equiconsistency results at the level of subcompact cardinals. Assuming SBHδ, a special case of the Strategic Branches Hypothesis, we prove that if δ is a Woodin cardinal and both 2(δ) and 2δ fail, then δ is subcompact in a class inner model. If in addition 2(δ) fails, we prove that δ is Π21 subcompact in a class inner model. These results are optimal, and lead to equiconsistencies. A...

Under the assumption that δ is a Woodin cardinal and GCH holds, I show that if F is any class function from the regular cardinals to the cardinals such that (1) κ < cf(F (κ)), (2) κ < λ implies F (κ) ≤ F (λ), and (3) δ is closed under F , then there is a cofinality-preserving forcing extension in which 2 = F (γ) for each regular cardinal γ < δ, and in which δ remains Woodin. Unlike the analogou...

In this paper, we explore the structure theory of L(R, μ) under the hypothesis L(R, μ) “AD + μ is a normal fine measure on Pω1(R)” and give some applications. First we show that “ZFC + there exist ω2 Woodin cardinals”1 has the same consistency strength as “AD + ω1 is R-supercompact”. During this process we show that if L(R, μ) AD then in fact L(R, μ) AD. Next we prove important properties of L(...

We prove the consistency (modulo supercompact) of a negative answer to the Cantor discontinuum partition problem (i.e., some Hausdorff compact space cannot be partitioned to two sets not containing a closed copy of Cantor discontinuum). In this model we have CH. Without CH we get consistency results using a pcf assumption, close relatives of which are necessary for such results; so we try to de...

The assumption of [HM] and our assumption here, the existence of a strong partition cardinal, is moderately special. On the one hand, it violates the Axiom of Choice and is not relatively consistent with ZF (unlike AC and its negation). On the other hand, under the Axiom of Determinacy (AD), such cardinals are abundant and consistent with countable choice and DC, the principle of Dependent Choi...

1. It is shown that the failure of ♦S , for a set S ⊆ אω+1 that reflects stationarily often, is consistent with GCH and APאω , relatively to the existence of a supercompact cardinal. By a theorem of Shelah, GCH and λ entails ♦S for any S ⊆ λ that reflects stationarily often. 2. We establish the consistency of existence of a stationary subset of [אω+1] that cannot be thinned out to a stationary ...

We present a characterization of supercompactness measures for ω1 in L(R), and of countable products of such measures, using inner models. We give two applications of this characterization, the first obtaining the consistency of δ13 = ω2 with ZFC+AD , and the second proving the uniqueness of the supercompactness measure over Pω1 (λ) in L(R) for λ > δ21. Starting with the work of Steel [13] it b...

We prove that the following two statements are equiconsistent: there exists a greatly Mahlo cardinal; there exists a regular uncountable cardinal κ such that no stationary subset of κ+ ∩ cof(κ) carries a partial square. A famous theorem in set theory is the result that the failure of the square principle κ, for a regular uncountable cardinal κ, is equiconsistent with a Mahlo cardinal. Solovay p...

We prove that if μ is a regular cardinal and P is a μ-centered forcing poset, then P forces that (I[μ++])V generates I[μ++] modulo clubs. Using this result, we construct models in which the approachability property fails at the successor of a singular cardinal. We also construct models in which the properties of being internally club and internally approachable are distinct for sets of size the...