The Filter Dichotomy says that every uniform nonmeager filter on the integers is mapped by a finite-to-one function to an ultrafilter. The consistency of this principle was proved by Blass and Laflamme. A medial limit is a universally measurable function from [Formula: see text] to the unit interval [0, 1] which is finitely additive for disjoint sets, and maps singletons to 0 and ω to 1. Christensen and Mokobodzki independently showed that the Continuum Hypothesis implies the existence of medial limits. (...) We show that the Filter Dichotomy implies that there are no medial limits. (shrink)

The forcing construction Pmax, invented by W. Hugh Woodin, produces a model whose collection of subsets of ω₁ is in some sense maximal. In this paper we study the Boolean algebra induced by the nonstationary ideal on ω₁ in this model. Among other things we show that the induced quotient does not have a simply definable form. We also prove several results about saturation properties of the ideal in this extension.

We shall show the consistency of CH+ᄀ(+) and CH+(+)+ there are no club guessing sequences on ω₁. We shall also prove that ◊⁺ does not imply the existence of a strong club guessing sequence ω₁.

The canonical function game is a game of length ω1 introduced by W. Hugh Woodin which falls inside a class of games known as Neeman games. Using large cardinals, we show that it is possible to force that the game is not determined. We also discuss the relationship between this result and Σ22 absoluteness, cardinality spectra and Π2 maximality for H(ω2) relative to the Continuum Hypothesis.

We present Woodin's proof that if there exists a measurable Woodin cardinal δ, then there is a forcing extension satisfying all $\Sigma _{2}^{2}$ sentences ϕ such that CH + ϕ holds in a forcing extension of V by a partial order in V δ . We also use some of the techniques from this proof to show that if there exists a stationary limit of stationary limits of Woodin cardinals, then in a homogeneous forcing extension there is an elementary embedding (...) j: V → M with critical point $\omega _{1}^{V}$ such that M is countably closed in the forcing extension. (shrink)

In this paper, we study some new examples of ideals on $$\omega $$ with maximal Tukey type (that is, maximal among partial orders of size continuum). This discussion segues into an examination of a refinement of the Tukey order—known as the weak Rudin–Keisler order—and its structure when restricted to these ideals of maximal Tukey type. Mirroring a result of Fremlin (Note Mat 11:177–214, 1991) on the Tukey order, we also show that there is an analytic P-ideal above all other analytic (...) P-ideals in the weak Rudin–Keisler order. (shrink)

In [P. Larson, Martin’s Maximum and the axiom , Ann. Pure App. Logic 106 135–149], we modified a coding device from [W.H. Woodin, The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, Walter de Gruyter & Co, Berlin, 1999] and the consistency proof of Martin’s Maximum from [M. Foreman, M. Magidor, S. Shelah, Martin’s Maximum. saturated ideals, and non-regular ultrafilters. Part I, Annal. Math. 127 1–47] to show that from a supercompact limit of supercompact cardinals one could force (...) Martin’s Maximum to hold while the axiom fails. Here we modify that argument to prove a stronger fact, that Martin’s Maximum is consistent with the existence of a wellordering of the reals definable in H without parameters, from the same large cardinal hypothesis. In doing so we give a much simpler proof of the original result. (shrink)

In his book on P max [7], Woodin presents a collection of partial orders whose extensions satisfy strong club guessing principles on ω | . In this paper we employ one of the techniques from this book to produce P max variations which separate various club guessing principles. The principle (+) and its variants are weak guessing principles which were first considered by the second author [4] while studying games of length ω | . It was shown in [1] that (...) the Continuum Hypothesis does not imply (+) and that (+) does not imply the existence of a club guessing sequence on ω | . In this paper we give an alternate proof of the second of these results, using Woodin's P max technology, showing that a strengthening of (+) does not imply a weakening of club guessing known as the Interval Hitting Principle. The main technique in this paper, in addition to the standard P m a x machinery, is the use of condensation principles to build suitable iterations. (shrink)

The stationary set splitting game is a game of perfect information of length ω1 between two players, unsplit and split, in which unsplit chooses stationarily many countable ordinals and split tries to continuously divide them into two stationary pieces. We show that it is possible in ZFC to force a winning strategy for either player, or for neither. This gives a new counterexample to Σ22 maximality with a predicate for the nonstationary ideal on ω1, and an example of a consistently (...) undetermined game of length ω1 with payoff de.nable in the second-order monadic logic of order. We also show that the determinacy of the game is consistent with Martin's Axiom but not Martin's Maximum. (shrink)

A function f from ω1 to the ordinals is called a canonical function for an ordinal α if f represents α in any generic ultrapower induced by forcing with math formula. We introduce here a method for coding sets of ordinals using canonical functions from ω1 to ω1. Combining this approach with arguments from, we show, assuming the Continuum Hypothesis, that for each cardinal κ there is a forcing construction preserving cardinalities and cofinalities forcing that every subset of κ is (...) an element of the inner model math formula. (shrink)

Argumentation as the public exchange of reasons is widely thought to enhance deliberative interactions that generate and justify reasonable public policies. Adopting an argumentation-theoretic perspective, we survey the norms that should govern public argumentation and address some of the complexities that scholarly treatments have identified. Our focus is on norms associated with the ideals of correctness and participation as sources of a politically legitimate deliberative outcome. In principle, both ideals are mutually coherent. If the information needed for a correct deliberative (...) outcome is distributed among agents, then maximising participation increases information diversity. But both ideals can also be in tension. If participants lack competence or are prone to biases, a correct deliberative outcome requires limiting participation. The central question for public argumentation, therefore, is how to strike a balance between both ideals. Rather than advocating a preferred normative framework, our main purpose is to illustrate the complexity of this theme. (shrink)

Psychologists' emerging interest in spirituality and religion as well as the relevance of each phenomenon to issues of psychological importance requires an understanding of the fundamental characteristics of each construct. On the basis of both historical considerations and a limited but growing empirical literature, we caution against viewing spirituality and religiousness as incompatible and suggest that the common tendency to polarize the terms simply as individual vs. institutional or ′good′ vs. ′bad′ is not fruitful for future research. Also cautioning against (...) the use of restrictive, narrow definitions or overly broad definitions that can rob either construct of its distinctive characteristics, we propose a set of criteria that recognizes the constructs' conceptual similarities and dissimilarities. Rather than trying to force new and likely unsuccessful definitions, we offer these criteria as benchmarks for judging the value of existing definitions. (shrink)

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)

At the forefront of international concerns about global legislation and regulation, a host of noted environmentalists and business ethicists examine ethical issues in consumption from the points of view of environmental sustainability, economic development, and free enterprise.

Samkhya is one of the oldest, if not the oldest, system of classical Indian philosophy. This book traces its history from the third or fourth century B. C. up through the twentieth century. The Encyclopedia as a whole will present the substance of the various Indian systems of thought to philosophers unable to read the Sanskrit and having difficulty in finding their way about in the translations (where such exist). This volume includes a lengthy introduction by Gerald James Larson, (...) which discusses the history of Samkhya and its philosophical contours overall. The remainder of the book includes summaries in English of all extant Sanskrit texts of the system. Originally published in 1987. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905. (shrink)

A study of the religion of Jesus in terms of its pagan and Jewish sources, its inner meaning and finally its redevelopment in the pagan world. Larson argues that the religion of the Essene Jesus was a grand "synthesis of human experience drawn from many cultures" and that this religion has been greatly distorted by the ritual of the Church.--F. E. B.

Countable homogeneous relational structures have been studied by many people. One area of focus is the Ramsey theory of such structures. After a review of background material, a partition theorem of Laflamme, Sauer, and Vuksanovic for countable homogeneous binary relational structures is discussed with a focus on the size of the set of unavoidable colors.

To the best of my knowledge, no one in recent decades has written a book of this magnitude about the semantics of natural language. Certainly, nothing available today matches this volume in depth, precision, and coherence. The authors present classical and recent results of linguistic semantics within the framework of interpretative T-theories and defend the philosophical foundations of their approach by showing how it fits into the larger enterprise of cognitive linguistics. The book also includes an array of excellent exercises (...) and a good list of references. (shrink)

Understanding how people comprehend visual narratives (including picture stories, comics, and film) requires the combination of traditionally separate theories that span the initial sensory and perceptual processing of complex visual scenes, the perception of events over time, and comprehension of narratives. Existing piecemeal approaches fail to capture the interplay between these levels of processing. Here, we propose the Scene Perception & Event Comprehension Theory (SPECT), as applied to visual narratives, which distinguishes between front-end and back-end cognitive processes. Front-end processes occur (...) during single eye fixations and are comprised of attentional selection and information extraction. Back-end processes occur across multiple fixations and support the construction of event models, which reflect understanding of what is happening now in a narrative (stored in working memory) and over the course of the entire narrative (stored in long-term episodic memory). We describe relationships between front- and back-end processes, and medium-specific differences that likely produce variation in front-end and back-end processes across media (e.g., picture stories vs. film). We describe several novel research questions derived from SPECT that we have explored. By addressing these questions, we provide greater insight into how attention, information extraction, and event model processes are dynamically coordinated to perceive and understand complex naturalistic visual events in narratives and the real world. (shrink)