The proposed heavy dependence on Pavlovian conditioning to account for social behavior confounds phylogenically and ontogenically selected behavior patterns and ignores the extension of the principle of selection by consequences from biological to learning theory. Instead of acknowledging operant relations, Domjan et al. construct vaguely specified mechanisms based upon anticipatory cost-benefit considerations that are not supported by the Pavlovian conditioning literature.

In lieu of an abstract, here is a brief excerpt of the content:Philosophy of Music Education Review 12.1 (2004) 89-93 [Access article in PDF] Richard Shusterman, Performing Live: Aesthetic Alternatives for the Ends of Art (New York: Cornell University Press, 2000) Performing Live can be ascribed to post-modern American pragmatism in its widest expression. The author's intention is to revalue aesthetic experience, as well as to expand its realm to the extent where such experience also encompasses areas alien to traditional (...) aesthetics (for example, popular art or the somatic arts of self-improvement), reaching "the art of living" and reclaiming life itself as performance.After the introduction, titled "Aesthetic Renewal for the Ends of Art: An Overture," the text is organized in two main parts: "Aesthetic Experience and Popular Art" and "Soma, Self, and Society." Along with its ambitious title, the book's organization and structure promise a consistency of approach and narrative that somewhat dissolves during the reading. Shusterman recognizes that the stylistic problems that arise from a compilation of articles spanning ten years, mixing newly written and revised articles, but appeals to the reader's appreciation of the "quality of the mix" (11). Despite this feature, he is a lucid and thoughtful communicator and his writing style is excellent.In order to propose new or revitalized alternatives for the ends of art, Shusterman [End Page 89] explores the crisis in traditional visions of art, pointing to the compartmentalization and lack of credibility in current artistic thought and practice. Hegel, Walter Benjamin, Gianni Vattimo, and Arthur Danto provide him with the philosophical arguments accounting for the modern "exhaustion" of art. He supports a dialectical approach, both naturalistic and historicist, and builds a solid case for the possibility of alternatives in aesthetic thought and experience. He proposes two alternatives in his introduction: The two most prominent sites for today's aesthetic alternatives are clearly the mass-media popular arts and the complex cluster of disciplines devoted to bodily beauty and the art of living as expressed in today's preoccupation with aesthetic lifestyles. (7) It is unclear why these alternatives are "clearly" "the two most prominent." The author does not build a case for this assumption or for the "newness" of these alternatives. The "compilation style" mentioned above and this lack of justification for one of the most original elements in the book make this text often read as a philosophical drafting board. Keeping these characteristics in mind and also critically pondering the degrees of success in the enterprise, the reader must give Shusterman credit for taking risks while suggesting concrete examples for an experiential, holistic philosophy. In this sense, he mentions (in a different context) that "all of us [are caught] between conceptual history and futuristic speculation" (145), and he takes to heart this situation and its challenges.The first chapter in Performing Live aims to redeem or revalue the concept of aesthetic experience. For Shusterman, the loss of "felt experience" in art makes the interest in this concept utterly relevant for art's survival. He traces the lineage of this aesthetic philosophy to John Dewey, Monroe Beardsley, Nelson Goodman, and Arthur Danto. Besides focusing on these authors, his analysis revisits some of the main currents of thought in Western philosophy on the topic (for example, Plato, Hume, Kant, Dickie, Adorno, Benjamin, Gadamer). He concludes that a pragmatic approach to aesthetics, applying the concept of aesthetic experience directionally (as a means towards the actual experience rather than as a definable concept), can provide new paths for revitalizing art. Throughout the text this concept is also used expansively. The post-modern justification of the proposed aesthetics, by allowing the extension of aesthetic experience to embrace "the art of living," can leave the reader wondering if Shusterman has not simply turned Kierkegaard's existentialist stages to self-realization upside down.The second chapter is an open defense of popular art, carried out by contesting the main complaints found in current art criticism. Shusterman's defense is not directed to the products of popular art, but against the claim that popular art is aesthetically "worthless." His definition of "popular art... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy of Music Education Review 12.1 (2004) 89-93 [Access article in PDF] Richard Shusterman, Performing Live: Aesthetic Alternatives for the Ends of Art (New York: Cornell University Press, 2000) Performing Live can be ascribed to post-modern American pragmatism in its widest expression. The author's intention is to revalue aesthetic experience, as well as to expand its realm to the extent where such experience also encompasses areas alien to traditional (...) aesthetics (for example, popular art or the somatic arts of self-improvement), reaching "the art of living" and reclaiming life itself as performance.After the introduction, titled "Aesthetic Renewal for the Ends of Art: An Overture," the text is organized in two main parts: "Aesthetic Experience and Popular Art" and "Soma, Self, and Society." Along with its ambitious title, the book's organization and structure promise a consistency of approach and narrative that somewhat dissolves during the reading. Shusterman recognizes that the stylistic problems that arise from a compilation of articles spanning ten years, mixing newly written and revised articles, but appeals to the reader's appreciation of the "quality of the mix" (11). Despite this feature, he is a lucid and thoughtful communicator and his writing style is excellent.In order to propose new or revitalized alternatives for the ends of art, Shusterman [End Page 89] explores the crisis in traditional visions of art, pointing to the compartmentalization and lack of credibility in current artistic thought and practice. Hegel, Walter Benjamin, Gianni Vattimo, and Arthur Danto provide him with the philosophical arguments accounting for the modern "exhaustion" of art. He supports a dialectical approach, both naturalistic and historicist, and builds a solid case for the possibility of alternatives in aesthetic thought and experience. He proposes two alternatives in his introduction: The two most prominent sites for today's aesthetic alternatives are clearly the mass-media popular arts and the complex cluster of disciplines devoted to bodily beauty and the art of living as expressed in today's preoccupation with aesthetic lifestyles. (7) It is unclear why these alternatives are "clearly" "the two most prominent." The author does not build a case for this assumption or for the "newness" of these alternatives. The "compilation style" mentioned above and this lack of justification for one of the most original elements in the book make this text often read as a philosophical drafting board. Keeping these characteristics in mind and also critically pondering the degrees of success in the enterprise, the reader must give Shusterman credit for taking risks while suggesting concrete examples for an experiential, holistic philosophy. In this sense, he mentions (in a different context) that "all of us [are caught] between conceptual history and futuristic speculation" (145), and he takes to heart this situation and its challenges.The first chapter in Performing Live aims to redeem or revalue the concept of aesthetic experience. For Shusterman, the loss of "felt experience" in art makes the interest in this concept utterly relevant for art's survival. He traces the lineage of this aesthetic philosophy to John Dewey, Monroe Beardsley, Nelson Goodman, and Arthur Danto. Besides focusing on these authors, his analysis revisits some of the main currents of thought in Western philosophy on the topic (for example, Plato, Hume, Kant, Dickie, Adorno, Benjamin, Gadamer). He concludes that a pragmatic approach to aesthetics, applying the concept of aesthetic experience directionally (as a means towards the actual experience rather than as a definable concept), can provide new paths for revitalizing art. Throughout the text this concept is also used expansively. The post-modern justification of the proposed aesthetics, by allowing the extension of aesthetic experience to embrace "the art of living," can leave the reader wondering if Shusterman has not simply turned Kierkegaard's existentialist stages to self-realization upside down.The second chapter is an open defense of popular art, carried out by contesting the main complaints found in current art criticism. Shusterman's defense is not directed to the products of popular art, but against the claim that popular art is aesthetically "worthless." His definition of "popular art... (shrink)

Reflection Principles are commonly thought to produce only strong axioms of infinity consistent with V = L. It would be desirable to have some notion of strong reflection to remedy this, and we have proposed Global Reflection Principles based on a somewhat Cantorian view of the universe. Such principles justify the kind of cardinals needed for, inter alia , Woodin’s Ω-Logic.

This volume explores how Catholicism began and continues to open its doors to the wider world and to other confessions in embracing ecumenism, thanks to the vision and legacy of the Second Vatican Council. It explores such themes as the twentieth century context preceding the council; parallels between Vatican II and previous councils; its distinctively pastoral character; the legacy of the council in relation to issues such as church-world dynamics, as well as to ethics, social justice, economic activity. Several chapters (...) discuss the role of women in the church before, during, and since the council. Others discern inculturation in relation to Vatican II. The book also contains a wide and original range of ecumenical considerations of the council, including by and in relation to Free Church, Reformed, Orthodox, and Anglican perspectives. Finally, it considers the Council’s ongoing promise and remaining challenges with regard to ecumenical issues, including a groundbreaking essay on the future of ecumenical dialogue by Cardinal Walter Kasper. (shrink)

Les trois dialogues composés par le cardinal Nicolas de Cues pendant l'été 1450 ne résument pas toute la pensée de cet auteur, mais ils éclairent d'un jour relativement nouveau sa réflexion sur le lien entre sagesse et savoir. Proche en cela des Anciens, Nicolas de Cues pense leur unité dans la lumière de l'Un - de la Déité, écrit-il parfois - réfléchie par la puissance de l'esprit humain. Cet esprit est compris comme imago dei, non pas image de Dieu, (...) car tout ce qui est image de Dieu, mais plutôt copie de Dieu, reprise de la toute-puissance divine dans les limites que lui impose, toutefois, le fait d'être finie. La vérité étant en elle-même inaccessible ici-bas - inattingible, écrit Nicolas de Cues - reste le développement de cette vérité ou de l'Un, c'est à dire ce monde que l'esprit a pour tâche de mesurer, de reprendre, de recréer. Savoir pour inventer un monde à venir, avec humilité et ouverture à l'Etranger : cela s'accorde précisément avec ce que certains historiens nomment l'Humanisme. (shrink)

This book assesses how Vatican II opened up the Catholic Church to encounter, dialogue, and engagement with other world religions. Opening with a contribution from the President of the Pontifical Council for Interreligious Dialogue, Cardinal Jean-Louis Tauran, it next explores the impact, relevance, and promise of the Declaration Nostra Aetate before turning to consider how Vatican II in general has influenced interfaith dialogue and the intellectual and comparative study of world religions in the postconciliar decades, as well as the (...) contribution of particular past and present thinkers to the formation of current interreligious and comparative theological methods. Additionally, chapters consider interreligious dialogue vis-à-vis theological anthropology in conciliar documents; openness to the spiritual practices of other faith traditions as a way of encouraging positive interreligious encounter; the role of lay and new ecclesial movements in interreligious dialogue; and the development of Monastic Interreligious Dialogue. Finally, it includes a range of perspectives on the fruits and future of Vatican’s II’s opening to particular faiths such as Judaism, Islam, Hinduism, and Buddhism. (shrink)

Quelles politiques linguistiques vont le mieux permettre de mettre en valeur les avantages du bilinguisme et l'apprentissage des langues tout en favorisant la pérennité des langues minoritaires à l'ère de la mondialisation? Cet article tente de répondre à la question en basant son propos sur l'exemple des politiques d'aménagement linguistique du Canada et du Québec. Il montre que la mondialisation linguistique exerce une pression supplémentaire sur un équilibre déjà fragile entre les francophones et les anglophones au Canada et il suggère (...) de réviser les principes actuels afin d'assurer la pérennité du français, notamment au Québec.What language policies will best help highlight the benefits of bilingualism and language learning while promoting the sustainability of minority languages ​​in the era of globalization? This article attempts to answer the question, basing his remarks on the example of language planning policies of Canada and Quebec. It shows that linguistic globalization puts additional pressure on already fragile balance between Francophones and Anglophones in Canada and suggested revising the current principles in order to ensure the sustainability of the French, especially in Quebec. (shrink)

Le terme 'sacrement' a été attribué à l'Église par Vatican II qui en avait retrouvé la signification grâce au récent renouveau de la patristique. Le 'mystère' spécifie traditionnellement tout sacrement. Le mystère fondateur de l'Église-communion en Christ invite aussi à interpréter son essence sans la réduire à n'être qu'une des voies possibles du salut, mais sans lui réserver non plus une exclusivité d'ordre 'physique'. L'Église est structurée selon sa mission de sainteté, sa finalité étant le salut du monde.

Résumé : L’actuel programme d’« Histoire et éducation à la citoyenneté » pour le 2e cycle du secondaire, portant sur l’histoire du Québec, a suscité depuis 2006 de nombreux commentaires dans les milieux intellectuels en raison de son contenu. Certains affirment que la transmission des grands événements historiques qui structurent la mémoire collective ou nationale y est inexistante, d’autres soutiennent qu’il doit favoriser le développement des compétences critiques et citoyennes, donc, l’autonomie de l’élève face à la mémoire collective ou nationale. (...) Quelle position les enseignants du secondaire occupent-ils dans ce débat ? Il est possible de le découvrir par l’entremise de la perception qu’ils se font du programme et de ses trois compétences. Abstract: The current program «History and Citizenship Education» for the second cycle of secondary education, on the Quebec history, raised many comments in intellectual circles because of its content since 2006. Some argue that the transmission of major historical events that shape the collective or national memory is non-existent, others argue that it should promote the development of critical thinking and citizenship skills, thus the autonomy of the student facing the memory collective or national. What position secondary teachers do they have in this debate? It’s findable out through the teachers perceptions about program and its three skills. (shrink)

Résumé : Cet article s’inscrit dans le cadre d’un programme de recherche portant sur les liens entre les missions d’instruction et de socialisation de l’école québécoise à travers l’enseignement de diverses matières scolaires. Les premiers résultats de la recherche ont montré que pour les enseignants, la construction de l’identité de l’élève a beaucoup plus d’importance que la construction de l’identité collective. Cependant, le processus de subjectivation identifié ne signifie pas pour autant qu’il n’y ait plus aucune transmission ou production de (...) « valeurs » ou de « références » communes dans l’enseignement des différentes matières. Par exemple, dans l’enseignement de l’histoire et éducation à la citoyenneté, la construction de l’identité individuelle et celle de l’identité collective sont en continuelle tension. Cependant, l’identité collective ne passe plus tant par le canal d’une mémoire identitaire ou d’un récit national que par la promotion d’une modernité articulée au pluralisme, à la critique, à la rationalité intersubjective et, paradoxalement, à la subjectivation précédemment évoquée. L’article portera donc particulièrement sur cette tendance observée au sein de l’enseignement du cours d’histoire et éducation à la citoyenneté. Abstract : This paper is part of a research program about the links between the missions of instruction and socialization of the Quebec school by the teaching of different school subjects. The first results of this research showed that for teachers, the construction of the identity of the student is much more important than the construction of collective identity. However, the process of subjectivation identified not mean that there is no more transmission or production of «values» or «references» common in teaching different school subjects. For example, in the teaching of history and citizenship education, construction of individual identity and the collective identity are in constant tension. However, the collective identity does not pass anymore by the way of a memory or a national story, but by the promotion of modernity articulated with pluralism, criticism, intersubjective rationality and, paradoxically, the subjectivation previously mentioned. This paper will therefore especially on this trend in the teaching of the course of history and citizenship education. (shrink)

A cardinal κ is tall if for every ordinal θ there is an embedding j: V → M with critical point κ such that j > θ and Mκ ⊆ M. Every strong cardinal is tall and every strongly compact cardinal is tall, but measurable cardinals are not necessarily tall. It is relatively consistent, however, that the least measurable cardinal is tall. Nevertheless, the existence of a tall cardinal is equiconsistent with the existence of a (...) strong cardinal. Any tall cardinal κ can be made indestructible by a variety of forcing notions, including forcing that pumps up the value of 2κ as high as desired. (shrink)

• We define a notion of order of indiscernibility type of a structure by analogy with Mitchell order on measures; we use this to define a hierarchy of strong axioms of infinity defined through normal filters, the α-weakly Erdős hierarchy. The filters in this hierarchy can be seen to be generated by sets of ordinals where these indiscernibility orders on structures dominate the canonical functions.• The limit axiom of this is that of greatly Erdős and we use it to calibrate (...) some strengthenings of the Chang property, one of which, CC+, is equiconsistent with a Ramsey cardinal, and implies that where K is the core model built with non-overlapping extenders — if it is rigid, and others which are a little weaker. As one corollary we have:TheoremIf then there is an inner model with a strong cardinal. • We define an α-Jónsson hierarchy to parallel the α-Ramsey hierarchy, and show that κ being α-Jónsson implies that it is α-Ramsey in the core model. (shrink)

The familiar Vendler-Kenny scheme of verb-types, viz., performances (further differentiated by Vedler into accomplishments and achievements), activities, and states, is too narrow in two important respects. First, it is narrow linguistically. It fails to take into account the phenomenon of verb aspect. The trichotomy is not one of verbs as lexical types but of predications. Second, the trichotomy is narrow ontologically. It is a specification in the context of human agency of the more fundamental, topic-neutral trichotomy, event-process-state.The central component in (...) this ontological trichotomy, event, can be sharply differentiated from its two flanking components by adapting a suggestion by Geoffrey N. Leech and others that the contrast between perfective and imperfective aspect in verbs corresponds to the count/mass distinction in the domain of nouns. With the help of two distinctions, of cardinal count adverbials versus frequency adverbials, and of occurrence versus associated occasion, two interrelated criteria for event predication are developed. Accordingly, Mary capsized the boat is an event predication because (a) it is equivalent to There was at least one capsizing of the boat by Mary, or (b) because it admits cardinal count adverbials, e.g., at least once, twice, three times. Ontologically speaking, events are defined as those occurrences that are inherently countable. (shrink)

Dans la septième dissertation de son Commentaire sur la République de Platon, Proclus fournit les éléments d’une philosophie politique néoplatonicienne très structurée. Fidèle, de façon générale, à la description platonicienne de l’âme tripartite et des quatre vertus cardinales, il introduit cependant d’importantes nuances dans cette théorie. L’idée de la prédominance d’une partie de l’âme sur une autre et l’idée de « vies mixtes » où deux parties de l’âme prédominent en même temps élargissent la description platonicienne des différents types poli­tiques. (...) En outre, en s’efforçant de donner diverses explications métaphysiques de la nature et du nombre des parties de l’âme et des vertus, Proclus inscrit la philosophie politique platonicienne plus ou moins tout d’une pièce dans une hiérarchie cosmique néoplatonicienne. (shrink)

Sortal predicates have been associated with a counting process, which acts as a criterion of identity for the individuals they correctly apply to. We discuss in what sense certain types of predicates suggested by quantum physics deserve the title of 'sortal' as well, although they do not characterize either a process of counting or a criterion of identity for the entities that fall under them. We call such predicates 'quantum-sortal predicates' and, instead of a process of counting, to them is (...) associated a 'criterion of cardinality'. After their general characterization, it is discussed how these predicates can be formally described. (shrink)

Let κ be a cardinal, and let H κ be the class of sets of hereditary cardinality less than κ ; let τ (κ) > κ be the height of the smallest transitive admissible set containing every element of {κ}∪H κ . We show that a ZFC-definable notion of long unfoldability, a generalisation of weak compactness, implies in the core model K, that the mouse order restricted to H κ is as long as τ. (It is known that some (...) weak large cardinal property is necessary for the latter to hold.) In other terms we delimit its strength as follows: Theorem Con(ZFC+ω2-Π 1 1-Determinacy) ⇒ ⇒Con(ZFC+V=K+∃ a long unfoldable cardinal ⇒ ⇒Con(ZFC+∀X(X # exists) + ‘‘ $\forall D \subseteq \omega_1 D$ is universally Baire ⇔ ∃r∈R(D∈L(r)))’’, and this is set-generically absolute). We isolate a notion of ω-closed cardinal which is weaker than an ω1-Erd\ os cardinal, and show that this bounds the first long unfoldable: Theorem Let κ be ω -closed. Then there is a long unfoldable ł<κ. (shrink)

This article traces the career path and personality of a chancery official or secretary in the service of the Gonzaga, the ruling dynasty of Mantua, in the middle years of the fifteenth century. It relates Giovanni Arrivabene to the contemporary social, political and cultural context of this secondary northern Italian power or signoria but touches the wider Italian world at many points, particularly the papal court, whether in Rome or other locations, where Giovanni’s talented younger brother served first as the (...) secretary of Cardinal Francesco Gonzaga and later as an apostolic secretary. The comparison between the two brothers and their interdependence as property owners is a recurring theme. Giovanni emerges with evident limitations which hampered the succession to his father’s superior role in the chancery, where other privileged families also tended to prevail. Humanist educated and efficient, he seems to have been a bureaucrat of routine abilities, not quite astute enough to outpace certain departmental rivals or to avoid being tainted by association with some possibly corrupt practice which caused loss of favour. His later rehabilitation, first as a provincial magistrate concerned with criminal jurisdiction and later as an estate management official, illustrates that private relationships within the narrow world of a civic quasi despotic government were all important. (shrink)

The interaction between large cardinals, determinacy of two-person perfect information games, and inner model theory has been a singularly powerful driving force in modern set theory during the last three decades. For the outsider the intellectual excitement is often tempered by the somewhat daunting technicalities, and the seeming length of study needed to understand the flow of ideas. The purpose of this article is to try and give a short, albeit rather rough, guide to the broad lines of development.

This book is intended for mathematicians. Its origins lie in a course of lectures given by an algebraist to a class which had just completed a sub stantial course on abstract algebra. Consequently, our treatment ofthe sub ject is algebraic. Although we assurne a reasonable level of sophistication in algebra, the text requires little more than the basic notions of group, ring, module, etc. A more detailed knowledge of algebra is required for some of . the exercises. We also assurne (...) a familiarity with the main ideas of set theory, including cardinal numbers and Zorn's Lemma. In this book, we carry out a mathematical study of the logic used in mathematics. We do this by constructing a mathematical model oflogic and applying mathematics to analyse the properties of the model. We therefore regard all our existing knowledge of mathematics as being applicable to the analysis of the model, and in particular we accept set theory as part of the meta-Ianguage. We are not attempting to construct a foundation on which all mathematics is to be based-rather, any conclusions to be drawn about the foundations of mathematics co me only by analogy with the model, and are to be regarded in much the same way as the conclusions drawn from any scientific theory. (shrink)

We use a reverse Easton forcing iteration to obtain a universe with a definable well-order, while preserving the GCH and proper classes of a variety of very large cardinals. This is achieved by coding using the principle ◊ $_{k^ - }^* $ at a proper class of cardinals k. By choosing the cardinals at which coding occurs sufficiently sparsely, we are able to lift the embeddings witnessing the large cardinal properties without having to meet any non-trivial master conditions.

We discuss saturating ultrafilters on N, relating them to other types of nonprincipal ultrafilter. (a) There is an (ω,c)-saturating ultrafilter on $\mathbf{N} \operatorname{iff} 2^\lambda \leq \mathfrak{c}$ for every $\lambda and there is no cover of R by fewer than c nowhere dense sets. (b) Assume Martin's axiom. Then, for any cardinal κ, a nonprincipal ultrafilter on N is (ω,κ)-saturating iff it is almost κ-good. In particular, (i) p(κ)-point ultrafilters are (ω,κ)-saturating, and (ii) the set of (ω,κ)-saturating ultrafilters is invariant (...) under homeomorphisms of $\beta\mathbf{N\backslash N}$ . (c) It is relatively consistent with ZFC to suppose that there is a Ramsey p(c)-point ultrafilter which is not (ω,c)-saturating. (shrink)

The Necessary Maximality Principle for c. c. c. forcing with real parameters is equiconsistent with the existence of a weakly compact cardinal. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim).

The consistency of a second-order version of Morley’s Theorem on the number of countable models was proved in [EHMT23] with the aid of large cardinals. We here dispense with them.

We show that if the weak compactness of a cardinal is made indestructible by means of any preparatory forcing of a certain general type, including any forcing naively resembling the Laver preparation, then the cardinal was originally supercompact. We then apply this theorem to show that the hypothesis of supercompactness is necessary for certain proof schemata.

We give limits defined in terms of abstract pointclasses of the amount of determinacy available in certain canonical inner models involving strong cardinals. We show for example: Theorem A. $\mathrm{D}\mathrm{e}\mathrm{t}\text{\hspace{0.17em}}({\mathrm{\Pi }}_{1}^{1}-\mathrm{I}\mathrm{N}\mathrm{D})$ ⇒ there exists an inner model with a strong cardinal. Theorem B. Det(AQI) ⇒ there exist type-1 mice and hence inner models with proper classes of strong cardinals. where ${\mathrm{\Pi }}_{1}^{1}-\mathrm{I}\mathrm{N}\mathrm{D}\phantom{\rule{0ex}{0ex}}$ (AQI) is the pointclass of boldface ${\mathrm{\Pi }}_{1}^{1}$ -inductive (respectively arithmetically quasi-inductive) sets of reals.

We present a new partial order for directly forcing morasses to exist that enjoys a significant homogeneity property. We then use this forcing in a reverse Easton iteration to obtain an extension universe with morasses at every regular uncountable cardinal, while preserving all n-superstrong , hyperstrong and 1-extendible cardinals. In the latter case, a preliminary forcing to make the GCH hold is required. Our forcing yields morasses that satisfy an extra property related to the homogeneity of the partial order; (...) we refer to them as mangroves and prove that their existence is equivalent to the existence of morasses. Finally, we exhibit a partial order that forces universal morasses to exist at every regular uncountable cardinal, and use this to show that universal morasses are consistent with n-superstrong, hyperstrong, and 1-extendible cardinals. This all contributes to the second author’s outer model programme, the aim of which is to show that L-like principles can hold in outer models which nevertheless contain large cardinals. (shrink)

We prove consistent, assuming there is a supercompact cardinal, that there is a singular strong limit cardinalμ, of cofinalityω, such that everyμ+-chromatic graphXonμ+has an edge colouringcofXintoμcolours for which every vertex colouringgofXinto at mostμmany colours has ag-colour class on whichctakes every value.The paper also contains some generalisations of the above statement in whichμ+is replaced by other cardinals >μ.

Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil (...) set theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science. (shrink)

The paper examines a single, apparently simple argument for the existence of incompatibilities between the virtues as traits of character. This argument appeals not to empirical truths about human psychology or human nature but to the possibility of conflict between the exercise of different virtues in action. There are, for example, situations in which we can exercise the virtue of truthfulness only at the expense of not exercising the virtue of tact, as when we are asked a question to which (...) the straightforward answer will pain our questioner. The person who in these situations would give a straightforward answer possesses the virtue of truthfulness to a higher degree than the person who would respond more evasively, whereas the latter person possesses the virtue of tact to a higher degree than the former. But since the two responses are mutually exclusive, it follows that the degree to which we possess the virtue of truthfulness must, at least in some measure, correlate inversely with the degree to which we possess the virtue of tact. And similarly for many other pairs of virtues. My paper focusses on the two cardinal assumptions of this argument, viz., (i) that sometimes the exercise of one virtue can conflict with the exercise of another, and (ii) that the degree to which a person possesses a virtue correlates directly with the extent of its exercise, and defends both assumptions against a variety of objections which allege that, in different ways, they are insensitive to the complexity of the virtues. (shrink)

We define a weak iterability notion that is sufficient for a number of arguments concerning $\Sigma _{1}$ -definability at uncountable regular cardinals. In particular we give its exact consistency strength first in terms of the second uniform indiscernible for bounded subsets of $\kappa $ : $u_2$, and secondly to give the consistency strength of a property of Lücke’s.TheoremThe following are equiconsistent:There exists $\kappa $ which is stably measurable;for some cardinal $\kappa $, $u_2=\sigma $ ;The $\boldsymbol {\Sigma }_{1}$ -club property (...) holds at a cardinal $\kappa $.Here $\sigma $ is the height of the smallest $M \prec _{\Sigma _{1}} H $ containing $\kappa +1$ and all of $H $. Let $\Phi $ be the assertion: TheoremAssume $\kappa $ is stably measurable. Then $\Phi $.And a form of converse:TheoremSuppose there is no sharp for an inner model with a strong cardinal. Then in the core model K we have: $\mbox {``}\exists \kappa \Phi \mbox {''}$ is -generically absolute ${\,\longleftrightarrow \,}$ There are arbitrarily large stably measurable cardinals.When $u_2 < \sigma $ we give some results on inner model reflection. (shrink)

We show that the predicate “x is the power set of y” is ‐definable, if V = L[E] is an extender model constructed from a coherent sequences of extenders, provided that there is no inner model with a Woodin cardinal. Here is a predicate true of just the infinite cardinals. From this we conclude: the validities of second order logic are reducible to, the set of validities of the Härtig quantifier logic. Further we show that if no L[E] model (...) has a cardinal strong up to one of its ℵ‐fixed points, and, the Löwenheim number of this logic, is less than the least weakly inaccessible δ, then (i) is a limit of measurable cardinals of K, and (ii) the Weak Covering Lemma holds at δ. (shrink)

A classic result of Baumgartner-Harrington-Kleinberg [1] implies that assuming CH a stationary subset of ω1 has a CUB subset in a cardinal-perserving generic extension of V, via a forcing of cardinality ω1. Therefore, assuming that $\omega_2^L$ is countable: { $X \in L \mid X \subseteq \omega_1^L$ and X has a CUB subset in a cardinal -preserving extension of L} is constructible, as it equals the set of constructible subsets of $\omega_1^L$ which in L are stationary. Is there a (...) similar such result for subsets of $\ omega_2^L$ ? Building on work of M. Stanley [9], we show that there is not. We shall also consider a number of related problems, examining the extent to which they are "solvable" in the above sense, as well as defining a notion of reduction between them. (shrink)

We obtain an algorithm to compute finite coproducts of finitely generated Gödel algebras, i.e. Heyting algebras satisfying the prelinearity axiom =1. We achieve this result using ordered partitions of finite sets as a key tool to investigate the category opposite to finitely generated Gödel algebras . We give two applications of our main result. We prove that finitely presented Gödel algebras have free products with amalgamation; and we easily obtain a recursive formula for the cardinality of the free Gödel algebra (...) over a finite number of generators first established by A. Horn. (shrink)

We lift Jensen's coding method into the context of Woodin cardinals. By a theorem of Woodin, any real which preserves a "strong witness" to Woodinness is set-generic. We show however that there are class-generic reals which are not set-generic but preserve Woodinness, using "weak witnesses".

We show that assuming modest large cardinals, there is a definable class of ordinals, closed and unbounded beneath every uncountable cardinal, so that for any closed and unbounded subclasses $P, Q, {\langle L[P],\in,P \rangle }$ and ${\langle L[Q],\in,Q \rangle }$ possess the same reals, satisfy the Generalised Continuum Hypothesis, and moreover are elementarily equivalent. Examples of such P are Card, the class of uncountable cardinals, I the uniform indiscernibles, or for any n the class $C^{n}{=_{{\operatorname {df}}}}\{ \lambda \, | (...) \, V_{\lambda } \prec _{{\Sigma }_{n}}V\}$ ; moreover the theory of such models is invariant under ZFC-preserving extensions. They also all have a rich structure satisfying many of the usual combinatorial principles and a definable wellorder of the reals. The inner model constructed using definability in the language augmented by the Härtig quantifier is thus also characterized. (shrink)