Ernest Schimmerling

Combinatorial principles in the core model for one Woodin cardinal

Abstract We study the fine structure of the core model for one Woodin cardinal, building of the work of Mitchell and Steel on inner models of the form L[ E → ] . We generalize to L[ E → ] some combinatorial principles that were shown by Jensen to hold in L. We show that L[ E → ] satisfies the statement: “□κ holds whenever κ ⩽ the least measurable cardinal λ of ◁ order λ++”. We introduce a hierarchy of combinatorial principles □κ, λ for 1 ⩽ λ ⩽ κ such that □ κ □ κ, 1 ⇒ □ κ, λ ⇒ □ κ, κ □ κ ∗ . We prove that if ( κ + ) v = (κ + ) L[ E → ] , then □κ,c∝(κ) holds in V. As an application, we show that ZFC + PFA ⇒ Con(ZFC + “there is a Woodin cardinal”) . We also obtain one Woodin cardinal as a lower bound on the consistency strength of stationary reflection at κ+ for a singular, countably closed limit cardinal κ such that (Vκ+)# exists; likewise for the failure of □ κ ∗ at such a κ.

Square in Core Models

We prove that in all Mitchell-Steel core models, □ k holds for all k . (See Theorem 2.) From this we obtain new consistency strength lower bounds for the failure of □ k if k is either singular and countably closed, weakly compact, or measurable. (Corollaries 5, 8, and 9.) Jensen introduced a large cardinal property that we call subcompactness ; it lies between superstrength and supercompactness in the large cardinal hierarchy. We prove that in all Jensen core models, □ k holds iff k is not subcompact. (See Theorem 15; the only if direction is essentially due to Jensen.)

CHARACTERIZATION OF □κ IN CORE MODELS

We present a general construction of a □κ-sequence in Jensens fine structural extender models. This construction yields a local definition of a canonical □κ-sequence as well as a characterization of those cardinals κ, for which the principle □κ fails. Such cardinals are called subcompact and can be described in terms of elementary embeddings. Our construction is carried out abstractly, making use only of a few fine structural properties of levels of the model, such as solidity and condensation.

The maximality of the core model

Our main results are: 1) every countably certified extender that coheres with the core model K is on the extender sequence of K, 2) K computes successors of weakly compact cardinals correctly, 3) every model on the maximal 1-small construction is an iterate of K, 4) (joint with W. J. Mitchell) K‖κ is universal for mice of height ≤ κ whenever κ ≥ א2, 5) if there is a κ such that κ is either a singular countably closed cardinal or a weakly compact cardinal, and <ω κ fails, then there are inner models with Woodin cardinals, and 6) an ω-Erdos cardinal suffices to develop the basic theory of K.

A Core Model Toolbox and Guide

The subject of this chapter is core model theory at the level where it involves iteration trees. Our toolbox includes a list of fundamental theorems that set theorists who are not necessarily core model theorists can use off the shelf in applications. It also includes a catalog of such applications. For those interested in the nuts and bolts of core model theory, we offer a guide to the monograph “The Core Model Iterability Problem” by John Steel. We also provide an outline of the paper “The covering lemma up to a Woodin cardinal” by William Mitchell, John Steel and Ernest Schimmerling. These two sections with proofs build on an initial segment of the Handbook of Set Theory chapter by John Steel, whereas the other three sections require only general knowledge of set theory.

A Finite Family Weak Square Principle

THEOREM 1.2. Suppose that L[E] is a core model.2 Assume that every countable premouse 4X which elementarily embeds into a level of L[E] is (col + 1 )-iterable. Then, for every A., 0<0 holds in L[E]. The minimal non-i-small mouse is essentially a sharp for an inner model with a Woodin cardinal. We originally proved Theorem 1.2 under the assumption that L[E] is 1-small, building on [MiSt] and [Sch2]. Some generalizations followed by combining our methods with those of [St2] and [SchSt2]. (For example, the tame countably certified core model Kc satisfies D<r.) In order to eliminate the smallness assumption all together, one replaces our use of the Dodd-Jensen lemma in proofs of condensation properties for L[E] with the weak Dodd-Jensen lemma of [NSt]. The proof that we present here will assume 1-smallness, however, so that we may honestly quote the results of [MiSt] and [Sch2, ?? 1-4].

Cardinal transfer properties in extender models

Abstract We prove that if L [ E ] is a Jensen extender model, then L [ E ] satisfies the Gap-1 morass principle. As a corollary to this and a theorem of Jensen, the model L [ E ] satisfies the Gap-2 Cardinal Transfer Property ( κ + + , κ ) → ( λ + + , λ ) for all infinite cardinals κ and λ .

Woodin cardinals, Shelah cardinals, and the Mitchell-Steel core model

Theorem 4 is a characterization of Woodin cardinals in terms of Skolem hulls and Mostowski collapses. We define weakly hyper-Woodin cardinals and hyper-Woodin cardinals. Theorem 5 is a covering theorem for the Mitchell-Steel core model, which is constructed using total background extenders. Roughly, Theorem 5 states that this core model correctly computes successors of hyper-Woodin cardinals. within the large cardinal hierarchy, in increasing order we have: measurable Woodin, weakly hyper-Woodin, Shelah, hyper-Woodin, and superstrong cardinals. (The comparison of Shelah versus hyper-Woodin is due to James Cummings.)

Some Calkin algebras have outer automorphisms

We consider various quotients of the C*-algebra of bounded operators on a nonseparable Hilbert space, and prove in some cases that, assuming some restriction of the Generalized Continuum Hypothesis, there are many outer automorphisms.

AN EQUICONSISTENCY RESULT ON PARTIAL SQUARES

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.