Since the work of Gödel and Cohen, which showed that Hilbert's First Problem was independent of the usual assumptions of mathematics, there have been a myriad of independence results in many areas of mathematics. These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond and square discovered by Jensen. Simultaneously, attempts have been made to find suitable (...) natural strengthenings of ZFC, primarily by Large Cardinal or Reflection Axioms. These two directions have tension between them in that Jensen's principles, which tend to suggest a rather rigid mathematical universe, are at odds with reflection properties. A third development was the discovery by Shelah of "PCF Theory", a generalization of cardinal arithmetic that is largely determined inside ZFC. In this paper we consider interactions between these three theories in the context of singular cardinals, focusing on the various implications between square and scales, and on consistency results between relatively strong forms of square and stationary set reflection. (shrink)

Since the work of Gödel and Cohen, which showed that Hilbert's First Problem was independent of the usual assumptions of mathematics, there have been a myriad of independence results in many areas of mathematics. These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond and square discovered by Jensen. Simultaneously, attempts have been made to find suitable (...) natural strengthenings of ZFC, primarily by Large Cardinal or Reflection Axioms. These two directions have tension between them in that Jensen's principles, which tend to suggest a rather rigid mathematical universe, are at odds with reflection properties. A third development was the discovery by Shelah of "PCF Theory", a generalization of cardinal arithmetic that is largely determined inside ZFC. In this paper we consider interactions between these three theories in the context of singular cardinals, focusing on the various implications between square and scales, and on consistency results between relatively strong forms of square and stationary set reflection. (shrink)

We present a survey of combinatorial set theory relevant to the study of singular cardinals and their successors. The topics covered include diamonds, squares, club guessing, forcing axioms, and PCF theory.

We start by studying the relationship between two invariants isolated by Shelah, the sets of good and approachable points. As part of our study of these invariants, we prove a form of “singular cardinal compactness” for Jensen's square principle. We then study the relationship between internally approachable and tight structures, which parallels to a certain extent the relationship between good and approachable points. In particular we characterise the tight structures in terms of PCF theory and use our characterisation to prove (...) some covering results for tight structures, along with some results on tightness and stationary reflection. Finally, we prove some absoluteness theorems in PCF theory, deduce a covering theorem, and apply that theorem to the study of precipitous ideals. (shrink)

We prove some consistency results about and δ, which are natural generalisations of the cardinal invariants of the continuum and . We also define invariants cl and δcl, and prove that almost always = cl and = cl.

We prove a number of consistency results complementary to the ZFC results from our paper [J. Cummings, M. Foreman, M. Magidor, Canonical structure in the universe of set theory: part one, Annals of Pure and Applied Logic 129 211–243]. We produce examples of non-tightly stationary mutually stationary sequences, sequences of cardinals on which every sequence of sets is mutually stationary, and mutually stationary sequences not concentrating on a fixed cofinality. We also give an alternative proof for the consistency of the (...) existence of stationarily many non-good points, show that diagonal Prikry forcing preserves certain stationary reflection properties, and study the relationship between some simultaneous reflection principles. Finally we show that the least cardinal where square fails can be the least inaccessible, and show that weak square is incompatible in a strong sense with generic supercompactness. (shrink)

. From a proper class of supercompact cardinals, we force and obtain a model in which the proper classes of strongly compact and strong cardinals precisely coincide. In this model, it is the case that no strongly compact cardinal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\kappa$\end{document} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $2^\kappa = \kappa^+$\end{document} supercompact.

Combining techniques of the first author and Shelah with ideas of Magidor, we show how to get a model in which, for fixed but arbitrary finite n, the first n strongly compact cardinals κ 1 ,..., κ n are so that κ i for i = 1,..., n is both the i th measurable cardinal and κ + i supercompact. This generalizes an unpublished theorem of Magidor and answers a question of Apter and Shelah.

Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing that any of their eight Boolean combinations can be forced to hold at${\kappa ^{ + + }}$, assuming that$\kappa = {\kappa ^{ < \kappa }}$and there is a weakly compact cardinal aboveκ.If in additionκis supercompact then we can forceκto be${\aleph _\omega }$in the extension. The proofs combine the techniques of adding and then destroying (...) a nonreflecting stationary set or a${\kappa ^{ + + }}$-Souslin tree, variants of Mitchell’s forcing to obtain the tree property, together with the Prikry-collapse poset for turning a large cardinal into${\aleph _\omega }$. (shrink)

Assuming that GCH holds and [Formula: see text] is [Formula: see text]-supercompact, we construct a generic extension [Formula: see text] of [Formula: see text] in which [Formula: see text] remains strongly inaccessible and [Formula: see text] for every infinite cardinal [Formula: see text]. In particular the rank-initial segment [Formula: see text] is a model of ZFC in which [Formula: see text] for every infinite cardinal [Formula: see text].

This paper explores the type of stakeholder engagement currently being undertaken by many organisations as part of social and ethical accounting, auditing and reporting processes. Specifically, the paper seeks to determine the extent to which current corporate practice iteratively promotes stakeholder participation in collaboratively designing accountability programmes, or whether it merely is a new term for canvassing stakeholder opinions. Arnstein’s Ladder of Citizen Participation is used as a conceptual model for positioning contemporary methods of stakeholder dialogue. The findings from interviews (...) with representatives from twenty‐nine UK and transnational organisations actively engaged in stakeholder dialogue are presented. This paper concludes that contemporary organisations seem to be seeking to engage with stakeholders in partnerships, yet their attempts still fall short of what Arnstein called Citizen Control, the level of achievement that non‐governmental organisations would like stakeholder engagement to attain. (shrink)

We prove a consistency result about square principles and stationary reflection which generalises the result of Ben-David and Magidor [4].

We use mixture of forcing and inner models techniques to get some results on the possible behaviours of the Mitchell ordering at a measurable к.

Monroe Eskew and \, 2016. https://mathoverflow.net/questions/217951/tree-properties-on-omega-1-and-omega-2) asked whether the tree property at \ implies there is no Kurepa tree. We prove that the tree property at \ is consistent with the existence of \-trees with as many branches as desired.

Background and Aims Progress towards validating amyloid beta as an early indicator of Alzheimer’s disease (AD) heightens the need for evaluation of stakeholders’ perspectives of the benefits and harms of preclinical testing in asymptomatic individuals. Methods Investigators conducted and analyzed 14 semi-structured interviews with family members of patients diagnosed with AD. Results Participants reported benefits, including the potential to seek treatment, make lifestyle changes, and prepare for cognitive impairment. Participants identified harms, including social harms, adverse life decisions, and psychological harms. (...) Nine participants reported either a “positive global perspective” or a “positive global perspective (qualified).” Conclusion Results from this study characterized stakeholders’ perspectives on the potential benefits and harms of clinical use of preclinical testing for AD. Investigators used data from this study to develop a framework that contributes to ongoing discussions that will evaluate widespread adoption of preclinical testing and will inform future research. (shrink)

We analyse the Mitchell ordering in a model where κ is P 2 κ-hypermeasurable and $2^{2^\kappa} > (2^\kappa)^+$.

We force and construct a model in which level by level equivalence between strong compactness and supercompactness holds, along with a strong form of diamond and a version of square consistent with supercompactness. This generalises a result due to the first author. There are no restrictions in our model on the structure of the class of supercompact cardinals.

We prove some results related to the problem of blowing up the power set of the least measurable cardinal. Our forcing results improve those of [1] by using the optimal hypothesis.

This volume contains the proceedings of the Logic at Harvard conference in honor of W. Hugh Woodin's 60th birthday, held March 27–29, 2015, at Harvard University. It presents a collection of papers related to the work of Woodin, who has been one of the leading figures in set theory since the early 1980s. The topics cover many of the areas central to Woodin's work, including large cardinals, determinacy, descriptive set theory and the continuum problem, as well as connections between set (...) theory and Banach spaces, recursion theory, and philosophy, each reflecting a period of Woodin's career. Other topics covered are forcing axioms, inner model theory, the partition calculus, and the theory of ultrafilters. This volume should make a suitable introduction to Woodin's work and the concerns which motivate it. The papers should be of interest to graduate students and researchers in both mathematics and philosophy of mathematics, particularly in set theory, foundations and related areas. (shrink)

We show that it is consistent with ZFC (relative to large cardinals) that every infinite Boolean algebra B has an irredundant subset A such that 2 |A| = 2 |B| . This implies in particular that B has 2 |B| subalgebras. We also discuss some more general problems about subalgebras and free subsets of an algebra. The result on the number of subalgebras in a Boolean algebra solves a question of Monk from [6]. The paper is intended to be accessible (...) as far as possible to a general audience, in particular we have confined the more technical material to a "black box" at the end. The proof involves a variation on Foreman and Woodin's model in which GCH fails everywhere. (shrink)

We attempt to make a connection between the sequences of measures used to define Radin forcing and the coherent sequences of extenders which are the basis of modern inner model theory. We show that in certain circumstances we can read off sequences of measures as defined by Radin from coherent sequences of extenders, and that we can define Radin forcing directly from a coherent extender sequence and a sequence of ordinals; this generalises Mitchell's construction of Radin forcing from a coherent (...) sequence of measures. (shrink)

It is obvious that ♦ implies the existence of an antichain of stationary sets of cardinality 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.

Jech and Shelah in J Symb Log, 55, 822–830 (1990) studied full reflection below ${\aleph_\omega}$ , and produced a model in which the extent of full reflection is maximal in a certain sense. We produce a model in which full reflection is maximised in a different direction.

We give upper and lower bounds for the consistency strength of the failure of a combinatorial principle introduced by Jensen. "Square on singular cardinals".

After September 11, Sullivan wrote that while he wasn’t worried about the heartland, “decadent coastal liberals may well mount a fifth column.†This in response, as is well known, to a thoughtful New Yorker essay by Susan Sontag. Sullivan, who Eric Alterman—not usually a sharp wordsmith—memorably calls “Young Roy Cohn†later issued “Sontag awards.†His attitude and his popularization of a sort of Lynne Cheneyist position on what “Views†are improper and thus should not be publicly aired, probably did far (...) more damage to civil liberties than Joe McCarthy’s pal Roy, who at the very least didn’t take himself seriously. (shrink)

We discuss the relationship between various weak distributive laws and games in Boolean algebras. In the first part we give some game characterizations for certain forms of Prikry’s “hyper-weak distributive laws”, and in the second part we construct Suslin algebras in which neither player wins a certain hyper-weak distributivity game. We conclude that in the constructible universe L, all the distributivity games considered in this paper may be undetermined.

We present an alternative proof that from large cardinals, we can force the tree property at $\kappa ^+$ and $\kappa ^{++}$ simultaneously for a singular strong limit cardinal $\kappa $. The advantage of our method is that the proof of the tree property at the double successor is simpler than in the existing literature. This new approach also works to establish the result for $\kappa =\aleph _{\omega ^2}$.

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 and stationary many sets that are organic but not tight. Working in the Constructible Universe, we give a characterization of organic and tight sets in terms of fine structure. 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 55–75] about the combinatorial principle Coherent Squares. (shrink)