Assume that there is no transitive class model of ZFC with a Woodin cardinal. Let nu be a singular ordinal such that nu > omega(2) and cf(nu) < |nu|. Suppose nu is a regular cardinal in K. Then nu is a measurable cardinal in K. Moreover, if cf(nu) > omega, then o(K )(nu) >= cf(nu).

In the 1970ies, Lev Bukovský proved a beautiful criterion for when V is a generic extension of a given inner model W , see [1] and [2]. Bukovský’s theorem recently served as a very useful tool in set theoretic geology, see e.g. [16] and [17], and also in inner model theoretic geology, see [9] and [11]. In this paper we shall give a proof of Bukovský’s theorem and a presentation of Woodin’s extender algebra (see e.g. [15, pp. 1657ff.]) in a uniform fashion – one argument and one forcing will produce both results, see Theorem 3.11. We shall also reproduce Usuba’s results on the set directedness of grounds and on the mantle of V in the presence of an extendible cardinal. We shall then discuss the application of these techniques to recent developments in inner model theoretic geology, namely to the theory of Varsovian models. This paper arose out of a series of talks which the author gave at the School of Mathematics of the IPM, Tehran (Iran), February 23-27, 2019. He would like to thank his host, Mohammad Golshani, for his exceptional hospitality during this visit. He would like to thank Ali Sadegh Daghighi for inviting him to write this paper. He also thanks Lev Bukovský, James Cummings, Mohammad Golshani, Joel

It is shown that if every real has a sharp and every subset of ω1 is constructible from a real, then there is an inner model with a Woodin cardinal. The following theorem was produced at the AIM meeting “Descriptive Inner Model Theory,” June 02–06, 2014, in a working group whose participants were the authors listed above. The authors would like to thank AIM for their generous hospitality. Theorem 0.1 Assume that every real has a sharp and every subset of ω1 is constructible from a real. Then there is an inner model with a Woodin cardinal. Let C denote the club filter on ω1, i.e., C = {X ⊂ ω1 : ∃C ⊂ ω1 club C ⊂ X}. Lemma 0.2 (Folklore) Assume that every real has a sharp and every subset of ω1 is constructible from a real. Then C is an ultrafilter. 1Cf. http://aimath.org/pastworkshops/innermodel.html

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.

Note to the instructor Acknowledgments 1. Preliminaries 2. ZFC 3. Order 4. Cardinality 5. Trees 6. Dense linear orderings 7. Filters and ideals Appendix. Summary of exercises on Boolean algebra Index.

Set theory is the mathematics of infinity and part of the core curriculum for mathematics majors. This book blends theory and connections with other parts of mathematics so that readers can understand the place of set theory within the wider context. Beginning with the theoretical fundamentals, the author proceeds to illustrate applications to topology, analysis and combinatorics, as well as to pure set theory. Concepts such as Boolean algebras, trees, games, dense linear orderings, ideals, filters and club and stationary sets are also developed. Pitched specifically at undergraduate students, the approach is neither esoteric nor encyclopedic. The author, an experienced instructor, includes motivating examples and over 100 exercises designed for homework assignments, reviews and exams. It is appropriate for undergraduates as a course textbook or for self-study. Graduate students and researchers will also find it useful as a refresher or to solidify their understanding of basic set theory.

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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.

Contributors Introduction 1. An introduction to Pmax forcing Paul B. Larson, Peter Lumsdaine and Yimu Yin 2. Countable Borel equivalence relations Simon Thomas and Scott Schneider 3. Set theory and operator algebras Ilijas Farah and Eric Wofsey 4. Set mapping reflection Justin Moore and David Milovich 5. An introduction to hyperlinear and sofic groups Vladimir G. Pestov and Aleksandra Kwiatkowska 6. Aronszajn trees and the SCH Itay Neeman and Spencer Unger 7. Iterated forcing and the continuum hypothesis Todd Eisworth, Justin Tatch Moore and David Milovich 8. Short extender forcing Moti Gitik and Spencer Unger 9. The complexity of classification problems in ergodic theory Alexander S. Kechris and Robin D. Tucker-Drob 10. On the strengths and weaknesses of weak squares Menachem Magidor and Chris Lambie-Hanson 11. Proper forcing remastered Boban Velickovic and Giorgio Venturi 12. Set theory and von Neumann algebras Asger Toernquist and Martino Lupini 13. The HOD dichotomy W. Hugh Woodin, Jacob Davis and Daniel Rodriguez.

We define organic sets and organically stationary sequences, which generalize tight sets and tightly stationary sequences respectively. We show that there are stationary many inorganic sets (Theorem 3) and stationary many sets that are organic but not tight (Theorem 4). Working in the Constructible Universe, we give a characterization of organic and tight sets in terms of fine structure (Theorem 7). We answer a related question posed in [J. Cummings, M. Foreman, M. Magidor, Canonical structure in the universe of set theory: Part two, Ann. Pure Appl. Logic 142 (2006) 55–75] about the combinatorial principle Coherent Squares (Corollary 9).

Abstract We prove new upper bound theorems on the consistency strengths of SPFA(θ), SPFA(θ-linked) and SPFA(θ+ -cc). Our results are in terms of (θ, Γ)-subcompactness, which is a new large cardinal notion that combines the ideas behind subcompactness and Γ-indescribability. Our upper bound for SPFA(ϲ-linked) has a corresponding lower bound, which is due to Neeman and appears in his follow-up to this paper. As a corollary, SPFA(ϲ-linked) and PFA(ϲ-linked) are each equiconsistent with the existence of a -indescribable cardinal. Our upper bound for SPFA(ϲ-c.c) is a -indescribable cardinal, which is consistent with V = L. Our upper bound for SPFA(ϲ+-linked) is a cardinals κ that is (κ+,)-subcompact, which is strictly weaker than κ+-supercompact. The axiom MM(ϲ) is a consequence of SPFA(ϲ+-linked) by a slight refinement of a theorem of Shelah. Our upper bound for SPFA(ϲ++-c.c.) is a cardinal κ that is (κ+, )-subcompact, which is also strictly weaker than κ+-supercompact.

Working in L(E), we examine which large cardinal properties ofimply that all stationary subsets of cof(<�)\ � + reflect.

We present a question about Suslin trees and the weak square hierarchy which was contributed to the list of open problems of the BIRS workshop.

We present a general construction of a □κ-sequence in Jensen's 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.

It is obvious that ♦ implies the existence of an antichain of stationary sets of cardinality Open image in new window which is the largest possible cardinality. We show that the obvious antichain is not maximal and find a less obvious extension of it by ℵ2 more stationary sets.

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We define what it means for a function on ω1 to be a collapsing function for λ and show that if there exists a collapsing function for (2  ω 1 )+, then there is no precipitous ideal on ω1. We show that a collapsing function for ω2 can be added by forcing. We define what it means to be a weakly ω1‐Erdös cardinal and show that in L[E], there is a collapsing function for λ iff λ is less than the least weakly ω1‐Erdös cardinal. As a corollary to our results and a theorem of Neeman, the existence of a Woodin limit of Woodin cardinals does not imply the existence of precipitous ideals on ω1. We also show that the following statements hold in L[E]. The least cardinal λ with the Chang property (λ, ω1) ↠ (ω1, ω) is equal to the least ω1‐Erdös cardinal. In particular, if j is a generic elementary embedding that arises from non‐stationary tower forcing up to a Woodin cardinal, then the minimum possible value of j(ω1) is the least ω1‐Erdös cardinal. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

We study some combinatorial principles intermediate between square and weak square. We construct models which distinguish various square principles, and show that a strengthened form of weak square holds in the Prikry model. Jensen proved that a large cardinal property slightly stronger than 1-extendibility is incompatible with square; we prove this is close to optimal by showing that 1-extendibility is compatible with square.