In this paper we consider whether L(R) has “enough information” to contain a counterexample to the continuum hypothesis. We believe this question provides deep insight into the difficulties surrounding the continuum hypothesis. We show sufficient conditions for L(R) not to contain such a counterexample. Along the way we establish many results about nonstationary towers, non-reflecting stationary sets, generalizations of proper and semiproper forcing and Chang's conjecture.

This paper attempts to present and organize several problems in the theory of Singular Cardinals. The most famous problems in the area (bounds for the ℶ-function at singular cardinals) are well known to all mathematicians with even a rudimentary interest in set theory. However, it is less well known that the combinatorics of singular cardinals is a thriving area with results and problems that do not depend on a solution of the Singular Cardinals Hypothesis. We present here an annotated collection (...) of representative problems with some references. Where the problems are novel, attribution is attempted and it is noted where money is attached to particular problems. Three closely related themes are represented in these problems: stationary sets and stationary set reflection, variations of square and approachability, and the singular cardinals hypothesis. Underlying many of them are ideas from Shelah's PCF theory. Important subthemes were mutual stationarity, Aronszajn trees, and superatomic Boolean Algebras. The author notes considerable overlap between this paper and the unpublished report submitted to the Banff Center for the Workshop on Singular Cardinals Combinatorics, May 1–5, 2004. (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)

It is shown in this paper that it is consistent (relative to almost huge cardinals) for various club guessing ideals to be saturated.

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)

In 1932, von Neumann proposed classifying the statistical behavior of differentiable systems. Joint work of B. Weiss and the author proved that the classification problem is complete analytic. Based on techniques in that proof, one is able to show that the collection of recursive diffeomorphisms of the 2-torus that are isomorphic to their inverses is $\Pi ^0_1$-hard via a computable 1-1 reduction. As a corollary there is a diffeomorphism that is isomorphic to its inverse if and only if the Riemann (...) Hypothesis holds, a different one that is isomorphic to its inverse if and only if Goldbach’s conjecture holds and so forth. Applying the reduction to the $\Pi ^0_1$-sentence expressing “ZFC is consistent” gives a diffeomorphism T of the 2-torus such that the question of whether $T\cong T^{-1}$ is independent of ZFC. (shrink)

If ω n has the tree property for all $2 \leq n and $2^{ , then for all X ∈ H ℵ ω and $n exists.

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)

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 introduce a natural principleStrong Chang Reflectionstrengthening the classical Chang Conjectures. This principle is between a huge and a two huge cardinal in consistency strength. In this note we prove that it implies the existence of an inner model with a huge cardinal. The technique we explore for building inner models with huge cardinals adapts to show thatdecisiveideals imply the existence of inner models with supercompact cardinals. Proofs for all of these claims can be found in [10].1,2.

This paper has two parts. The first is concerned with a variant of a family of games introduced by Holy and Schlicht, that we call Welch games. Player II having a winning strategy in the Welch game of length [Formula: see text] on [Formula: see text] is equivalent to weak compactness. Winning the game of length [Formula: see text] is equivalent to [Formula: see text] being measurable. We show that for games of intermediate length [Formula: see text], II winning implies (...) the existence of precipitous ideals with [Formula: see text]-closed, [Formula: see text]-dense trees. The second part shows the first is not vacuous. For each [Formula: see text] between [Formula: see text] and [Formula: see text], it gives a model where II wins the games of length [Formula: see text], but not [Formula: see text]. The technique also gives models where for all [Formula: see text] there are [Formula: see text]-complete, normal, [Formula: see text]-distributive ideals having dense sets that are [Formula: see text]-closed, but not [Formula: see text]-closed. (shrink)

If $\omega_n$ has the tree property for all $2 \leq n < \omega$ and $2^{<\aleph_{\omega}} = \aleph_{\omega}$, then for all $X \in H_{\aleph_{\omega}}$ and $n < \omega, M^#_n$ exists.

The Ethical Journalist gives aspiring journalists the tools they need to make responsible professional decisions. Provides a foundation in applied ethics in journalism Examines the subject areas where ethical questions most frequently arise in modern practice Incorporates the views of distinguished print, broadcast and online journalists, exploring such critical issues as race, sex, and the digitalization of news sources Illustrated with 24 real-life case studies that demonstrate how to think in 'shades of gray' rather than 'black and white' Includes questions (...) for class discussion and guides for putting important ethical concepts to use in the real world Accompanying website includes model course schedules, discussion guides, PowerPoint slides, sample quiz and exam questions and links to additional readings online: www.wiley.com/go/foreman. (shrink)

MacIntyre argues that management embodies emotivism, and thus is inherently amoral and manipulative. His claim that management is necessarily Weberian is, at best, outdated, and the notion that management aims to be neutral and value free is incorrect. However, new forms of management, and in particular the increased emphasis on leadership which emerged after MacIntyre’s critique was published, tend to support his central charge. Indeed, charismatic and transformational forms of leadership seem to embody emotivism to a greater degree than do (...) more Weberian, bureaucratic forms of management; hence, MacIntyre’s central contention about our emotivistic culture seems to be well founded. Having criticised the details but defended the essence of MacIntyre’s critique of management, this paper sketches a MacIntyrean approach to management and leadership by highlighting the affinities between MacIntyre’s political philosophy and Greenleaf’s concept of servant leadership. (shrink)

Beneath the stylized violence and thrilling car crashes, the Mad Max films consider universal questions about the nature of human life, order and anarchy, justice and moral responsibility, society and technology, and ultimately, human redemption. In Mad Max and Philosophy, a diverse team of political scientists, historians, and philosophers investigates the underlying themes of the blockbuster movie franchise, following Max as he attempts to rebuild himself and the world. -/- This book guides you through the barren wastelands of a post-apocalyptic (...) future as you explore ethics and politics in The Wasteland, the importance of costumes and music, humankind's relationship with nature, commerce, gender, religion, madness, and more. (shrink)

The first edition of The Scientific Background to Modern Philosophy took the dialogue of science and philosophy from Aristotle through to Newton. This second edition adds eight chapters, taking the dialogue through the Enlightenment and up to Darwin. This anthology is an attempt to help bridge the gap between the history of science and the history of philosophy.

This paper initiates analytic inquiry into the metaphysics of masculinity. I argue that individual masculinities (such as ‘clone masculinity’ and ‘incel masculinity’) are distinct homeostatic property cluster kinds related to gender structures via processes of adherence, failed-adherence, selective adherence, and/or reinterpretation with respect to male-coded social norms.