In lieu of an abstract, here is a brief excerpt of the content:THE IMPACT OF VER/TATIS SPLENDOR ON CATHOLIC EDUCATION AT THE UNIVERSITY AND SECONDARY LEVELS* CARDINAL PIO LAGHI Prefect of the Sacred Congregationfor Catholic Education INTRODUCTION T HE TOPIC which has been proposed to me, "The Impact of Veritatis Splendor on Catholic Education at the University and Secondary Levels,'' requires a note of clarification with regard to the word impact. When this Encyclical Letter of Pope John Paul II (...) appeared, it was received with much comment, both positive and negative, on the part of the press and the other media. This is one aspect of the topic proposed : to trace in a synthetic way a picture of the reactions which the document stirred. Such would certainly be an interesting study, but I would perhaps be led to traverse much terrain before arriving at the nucleus of what I feel is my duty to say to you. I shall limit myself, therefore, to commenting on two positive reactions appearing in Italian newspapers immediately after the publication of the Encyclical. The first was in the well-known Roman newspaper, Il Tempo, under the title, "An Act of Consistency in an Epoch of Doubt." It states: "If lighthouses had been moved every month, the sailors in the night would have seen the ships of history dashed against the rocks; and no voyager would have reached his homeland again if the North Star, within the Zodiac, had to obey vacillating positions of fashions and ideologies.... The papal document stirs discussions.... Nevertheless it is very much of current interest. While many ships finish on the rocks because they have * This paper was originally presented at the Forum on Catholic University Education at Duquesne University in Pittsburgh, 3 November 1995. 1 2 CARDINAL PIO LAGHI sailed following fireflies instead of lighthouses and while the ruling classes have shipwrecked for having chosen compasses that lie, there is abroad a need to see the permanence of certain values recognized in the midst of all that changes, the necessity of an ethical nucleus linked to principles of the human person, honesty, freedom and responsibility." This is a positive reaction, then, which welcomes the Encyclical as a text which says things that have to be said, giving basic directions for a journey, and stating the things no one dares to speak but of which everyone knows the necessity in daily living. This is also the substance of another comment on the Encyclical that I think it would be useful to cite. Carlo Bo, a university professor and a well-known figure in Italian intellectual life, writes: "Modern culture no longer has objective criteria for distinguishing good from evil,'' and he adds, "John Paul II has fought against Communism in defense of freedom; now he criticizes Western culture, which makes of freedom an absolute value.... The exaltation of freedom leads to ethical relativism, against which the Pope raises his voice." Agreement and basic approval are transparent in these lines, but the title under which they appear is "A Call to Order." Immediately the image is evoked-negative to the sensitivities of our times-of a commander intent on imposing order on his soldiers. The heart of the problem perhaps, at least with regard to the teaching of moral theology in the Catholic context, is in the clear perception of the need for an unchanging point of reference, of a lighthouse for the voyage, and at the same time the fear of speaking of this need, of making it the subject of reflection, communication, and teaching. There is a kind of widespread fear which keeps us at a distance from the truth, from the permanent foundations of human acting, from the objective and the universal; a fear which is perhaps the principal cause on account of which so great a part of contemporary humanity risks dying of thirst while standing before a spring of cool water. In order not to appear authoritarian or negative, one keeps silent, does not speak, expresses oneself in a partial or even an erroneous way with regard to all that is most necessary for life. It would be difficult not to see in this a problem which is typically an educa- THE IMPACT OF VERITATIS SPLENDOR 3... (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)

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)

The aim of this work was to compare experimental investigations on effects of lidocaine, calcium and, BRL 34915 on reentries to simulated data obtained by use of a model of propagation based on the Huygens' constriction method already described in previous works. Calcium and lidocaine effects are investigated on anisotropic conduction conditions. In both cases, reduction in conduction velocities are observed. In lidocaine case, a refractory area is located along the longitudinal axis. In agreement with experimental electrical mapping, the simulations (...) show that the stabilization of reentrant excitation is mainly due to the existence of this refractory area around which the reentrant circuit can develop. The experimental study shows that BRL 34915 has both arrhythmogenic and antiarrhythmic effects. A detailed electrophysiological analysis has shown that drug infusion act on normal cardiac cells by decreasing the relative and absolute refractory period. BRL 34915 action is simulated by a decrease in the refractory period showing that the time frequency of the reentrant activity is increased and that the spatial size where the reentry is developing is becoming smaller. These two effects are arrhythmogenic, the simulated data being so in good agreement with the experimental ones. (shrink)

We study cardinal invariants connected to certain classical orderings on the family of ideals on ω. We give topological and analytic characterizations of these invariants using the idealized version of Fréchet-Urysohn property and, in a special case, using sequential properties of the space of finitely-supported probability measures with the weak* topology. We investigate consistency of some inequalities between these invariants and classical ones, and other related combinatorial questions. At last, we discuss maximality properties of almost disjoint families related to (...) certain ordering on ideals. (shrink)

We introduce a natural two-cardinal version of Bagaria’s sequence of derived topologies on ordinals. We prove that for our sequence of two-cardinal derived topologies, limit points of sets can be characterized in terms of a new iterated form of pairwise simultaneous reflection of certain kinds of stationary sets, the first few instances of which are often equivalent to notions related to strong stationarity, which has been studied previously in the context of strongly normal ideals. The non-discreteness of these (...) two-cardinal derived topologies can be obtained from certain two-cardinal indescribability hypotheses, which follow from local instances of supercompactness. Additionally, we answer several questions posed by the first author, Holy and White on the relationship between Ramseyness and indescribability in both the cardinal context and in the two-cardinal context. (shrink)

Different analyses of complex cardinal numerals have been proposed in Generative Grammar. This article provides an analysis of these expressions based on the Strong Minimalist Thesis, according to which the derivations of linguistic expressions are generated by a simple combinatorial operation, applying in accord with principles external to the language faculty. The proposed derivations account for the asymmetrical structure of additive and multiplicative complexes and for the instructions they provide to the external systems for their interpretation. They harmonize with (...) those of coordinate nouns, and thus offer a unified Minimalist account of their core properties. Firstly, the empirical problem addressed is stated. Secondly, the theoretical framework is presented. Thirdly, Minimalist derivations for additive and multiplicative complexes are provided. Fourthly, the proposed derivations are contrasted with derivations not relying on the Strong Minimalist Thesis. Lastly, consequences for linguistic theory are identified as well as questions open to further inquiry. (shrink)

Quasi-set theory is a ZFU-like axiomatic set theory, which deals with two kinds of ur-elements: M-atoms, objects like the atoms of ZFU, and m-atoms, items for which the usual identity relation is not defined. One of the motivations to advance such a theory is to deal properly with collections of items like particles in non-relativistic quantum mechanics when these are understood as being non-individuals in the sense that they may be indistinguishable although identity does not apply to them. According to (...) some authors, this is the best way to understand quantum objects. The fact that identity is not defined for m-atoms raises a technical difficulty: it seems impossible to follow the usual procedures to define the cardinal of collections involving these items. In this paper we propose a definition of finite cardinals in quasi-set theory which works for collections involving m-atoms. (shrink)

There are multiple formal characterizations of the natural numbers available. Despite being inter-derivable, they plausibly codify different possible applications of the naturals – doing basic arithmetic, counting, and ordering – as well as different philosophical conceptions of those numbers: structuralist, cardinal, and ordinal. Some influential philosophers of mathematics have argued for a non-egalitarian attitude according to which one of those characterizations is ‘more basic’ or ‘more fundamental’ than the others. This paper addresses two related issues. First, we review some (...) of these non-egalitarian arguments, lay out a laundry list of different, legitimate, notions of relative priority, and suggest that these arguments plausibly employ different such notions. Secondly, we argue that given a metaphysical-cum-epistemological gloss suggested by Frege's foundationalist epistemology, the ordinals are plausibly more basic than the cardinals. This is just one orientation to relative priority one could take, however. Ultimately, we subscribe to an egalitarian attitude towards these formal characterizations: they are, in some sense, equally ‘legitimate’. (shrink)

One of the numerous characterizations of a Ramsey cardinal κ involves the existence of certain types of elementary embeddings for transitive sets of size κ satisfying a large fragment of ZFC. We introduce new large cardinal axioms generalizing the Ramsey elementary embeddings characterization and show that they form a natural hierarchy between weakly compact cardinals and measurable cardinals. These new axioms serve to further our knowledge about the elementary embedding properties of smaller large cardinals, in particular those still (...) consistent with V = L. (shrink)

The cardinal role that notions of respect and self-respect play in Rawls’s A Theory of Justice has already been abundantly examined in the literature. However, it has hardly been noticed that these notions are also central for Michael Walzer’s Spheres of Justice. Respect and self-respect are not only central topics of his chapter on “recognition”, but constitute a central aim of his whole theory of justice. This paper substantiates this thesis and elucidates Walzer’s criticism of Rawls’s that we need (...) to distinguish between “self-respect” and “self-esteem”. (shrink)

Unfoldable cardinals are preserved by fast function forcing and the Laver-like preparations that fast functions support. These iterations show, by set-forcing over any model of ZFC, that any given unfoldable cardinal κ can be made indestructible by the forcing to add any number of Cohen subsets to κ.

ROLFE G. Nursing Inquiry 2012; 19: 98–106 [Epub ahead of print]Cardinal John Henry Newman and ‘the ideal state and purpose of a university’: nurse education, research and practice development for the twenty‐first centuryCardinal John Henry Newman’s book, The Idea of a University, first published in the mid nineteenth century, is often invoked as the epitome of the liberal Enlightenment University in discussions and debates about the role and purpose of nurse education. In this article I will examine Newman’s book (...) in greater detail and with a more critical eye than is generally the case in the writing of nurse academics. In particular, I will focus on the claims that Newman was a champion of the Enlightenment University of the nineteenth century, that he promoted the idea of ‘disinterested’ universal knowledge for its own sake, that he was an early advocate of the pursuit of knowledge through scientific research, and the supposition that he would have welcomed the discipline of nursing into the University. In each case, I will suggest that these claims are based on an extremely selective reading of Newman’s work. I will conclude by employing the example of practice development to propose an alternative way for nursing to find its place in the modern University that does not involve a retreat into what I will argue is an outdated and nostalgic view of the aims and purpose of higher education. (shrink)

The cardinal role that notions of respect and self-respect play in Rawls’s A Theory of Justice has already been abundantly examined in the literature. In contrast, it has hardly been noticed that these notions are also central to Michael Walzer’s Spheres of Justice. Respect and self-respect are not only central topics of his chapter “Recognition”, but constitute a central aim of a “complex egalitarian society” and of Walzer’s theory of justice. This paper substantiates this thesis and elucidates Walzer’s criticism (...) of Rawls that we need to distinguish between “self-respect” and “self-esteem”. (shrink)

Unfoldable cardinals are preserved by fast function forcing and the Laver-like preparations that fast functions support. These iterations show, by set-forcing over any model of ZFC, that any given unfoldable cardinal $\kappa$ can be made indestructible by the forcing to add any number of Cohen subsets to $\kappa$.

There are multiple formal characterizations of the natural numbers available. Despite being inter-derivable, they plausibly codify different possible applications of the naturals – doing basic arithmetic, counting, and ordering – as well as different philosophical conceptions of those numbers: structuralist, cardinal, and ordinal. Nevertheless, some influential philosophers of mathematics have argued for a non-egalitarian attitude according to which one of those characterizations is more “legitmate” in virtue of being “more basic” or “more fundamental”. This paper addresses two related issues. (...) First, we review some of these non-egalitarian arguments, lay out a laundry list of different, legitimate, notions of relative priority, and suggest that these arguments plausibly employ different such notions. Secondly, we argue that given a metaphysical-cum-epistemological gloss suggested by Frege’s foundationalist epistomology, the ordinals are plausibly more basic than the cardinals. This is just one orientation to relative priority one could take, however. Ultimately, we subscribe to an egalitarian attitude towards these formal characterizations: they are, in some sense, equally “legitimate”. (shrink)

We reexamine some of the classic problems connected with the use of cardinal utility functions in decision theory, and discuss Patrick Suppes's contributions to this field in light of a reinterpretation we propose for these problems. We analytically decompose the doctrine of ordinalism, which only accepts ordinal utility functions, and distinguish between several doctrines of cardinalism, depending on what components of ordinalism they specifically reject. We identify Suppes's doctrine with the major deviation from ordinalism that conceives of utility functions (...) as representing preference differences, while being non- etheless empirically related to choices. We highlight the originality, promises and limits of this choice-based cardinalism. (shrink)

We show that many large cardinal notions up to measurability can be characterized through the existence of certain filters for small models of set theory. This correspondence will allow us to obtain a canonical way in which to assign ideals to many large cardinal notions. This assignment coincides with classical large cardinal ideals whenever such ideals had been defined before. Moreover, in many important cases, relations between these ideals reflect the ordering of the corresponding large cardinal (...) properties both under direct implication and consistency strength. (shrink)

The HOD Dichotomy Theorem states that if there is an extendible cardinal, δ, then either HOD is “close” to V or HOD is “far” from V. The question is whether the future will lead to the first or the second side of the dichotomy. Is HOD “close” to V, or “far” from V? There is a program aimed at establishing the first alternative—the “close” side of the HOD Dichotomy. This is the program of inner model theory. In recent years (...) the third author has provided evidence that there is an ultimate inner model—Ultimate-L—and he has isolated a natural conjecture associated with the model—the Ultimate-L Conjecture. This conjecture implies that that the first alternative holds—HOD is “close” to V. This is the future in which pattern prevails. In this paper we introduce a very different program, one aimed at establishing the second alternative—the “far” side of the HOD Dichotomy. This is the program of large cardinals beyond choice. Kunen famously showed that if AC holds then there cannot be a Reinhardt cardinal. It has remained open whether Reinhardt cardinals are consistent in ZF alone. It turns out that there is an entire hierarchy of choiceless large cardinals of which Reinhardt cardinals are only the beginning, and, surprisingly, this hierarchy appears to be highly ordered and amenable to systematic investigation, as we shall show in this paper. The point is that if these choiceless large cardinals are consistent then the Ultimate-L Conjecture must fail. This is the future where chaos prevails. (shrink)

A cardinal characteristic can often be described as the smallest size of a family of sequences which has a given property. Instead of this traditional concern for a smallest realization of the given property, a basically new approach, taken in [4] and [5], asks for a realization whose members are sequences of labels that correspond to 1-way infinite paths in a labelled graph. We study this approach as such, establishing tools that are applicable to all these cardinal characteristics. (...) As an application, we demonstrate the power of the tools developed by presenting a short proof of the bounded graph conjecture [4]. (shrink)

We construct models for the level-by-level equivalence between strong compactness and supercompactness containing failures of the Generalized Continuum Hypothesis at inaccessible cardinals. In one of these models, no cardinal is supercompact up to an inaccessible cardinal, and for every inaccessible cardinal $\delta $, $2^{\delta }\gt \delta ^{++}$. In another of these models, no cardinal is supercompact up to an inaccessible cardinal, and the only inaccessible cardinals at which GCH holds are also measurable. These results extend (...) and generalize earlier work of the author. (shrink)

We provide a model theoretical and tree property-like characterization of $\lambda $ - $\Pi ^1_1$ -subcompactness and supercompactness. We explore the behavior of these combinatorial principles at accessible cardinals.

A metric space (X, d) is monotone if there is a linear order < on X and a constant c such that d(x, y) ≤ c d(x, z) for all x < y < z in X. We investigate cardinal invariants of the σ-ideal Mon generated by monotone subsets of the plane. Since there is a strong connection between monotone sets in the plane and porous subsets of the line, plane and the Cantor set, cardinal invariants of these (...) ideals are also investigated. In particular, we show that non(Mon) ≥ ������ σ-linked , but non(Mon) < ������ σ-centered is consistent. Also cov(Mon) < c and cof(������) < cov(Mon) are consistent. (shrink)

We prove that in many situations it is consistent with ZFC that part of the invariants involved in Cichon's diagram are equal to κ while the others are equal to λ, where $\kappa < \lambda$ are both arbitrary regular uncountable cardinals. We extend some of these results to the case when λ is singular. We also show that $\mathrm{cf}(\kappa_U(\mathscr{L})) < \kappa_A(\mathscr{M})$ is consistent with ZFC.

We prove that in many situations it is consistent with ZFC that part of the invariants involved in Cichon's diagram are equal to $\kappa$ while the others are equal to $\lambda$, where $\kappa < \lambda$ are both arbitrary regular uncountable cardinals. We extend some of these results to the case when $\lambda$ is singular. We also show that $\mathrm{cf}) < \kappa_A$ is consistent with ZFC.

Models of set theory are constructed where the non-stationary ideal on PΩ1λ is presaturated. The initial model has a Woodin cardinal. Using the Lévy collapse the Woodin cardinal becomes λ+ in the final model. These models provide new information about the consistency strength of a presaturated ideal onPΩ1λ for λ greater than Ω1.

I provide indestructibility results for large cardinals consistent with V = L, such as weakly compact, indescribable and strongly unfoldable cardinals. The Main Theorem shows that any strongly unfoldable cardinal κ can be made indestructible by <κ-closed. κ-proper forcing. This class of posets includes for instance all <κ-closed posets that are either κ -c.c, or ≤κ-strategically closed as well as finite iterations of such posets. Since strongly unfoldable cardinals strengthen both indescribable and weakly compact cardinals, the Main Theorem therefore (...) makes these two large cardinal notions similarly indestructible. Finally. I apply the Main Theorem forcing extension preserving all strongly unfoldable cardinals in which every strongly unfoldable cardinal κ is indestructible by <κ-closed. κ-proper forcing. (shrink)

We study the influence of the existence of large cardinals on the existence of wellorderings of power sets of infinite cardinals κ with the property that the collection of all initial segments of the wellordering is definable by a Σ1‐formula with parameter κ. A short argument shows that the existence of a measurable cardinal δ implies that such wellorderings do not exist at δ‐inaccessible cardinals of cofinality not equal to δ and their successors. In contrast, our main result shows (...) that these wellorderings exist at all other uncountable cardinals in the minimal model containing a measurable cardinal. In addition, we show that measurability is the smallest large cardinal property that imposes restrictions on the existence of such wellorderings at uncountable cardinals. Finally, we generalise the above result to the minimal model containing two measurable cardinals. (shrink)

The first international Newman Conference took place in Luxembourg from July 23rd till July 28th, 1956. Many representatives of various countries of Europe were present. Apart from the well nigh perfect organization, it was undoubtedly this common sympathy and admiration for Cardinal Newman that explains the extremely pleasant atmosphere that prevailed throughout these days. It was not, however, a dolce far niente, because every day was filled with addresses and lectures which in various ways and from differing points of (...) view tried to estimate the importance of Newman for the modern age. Some lecturers looked back into the history of the Cardinal and showed how in fact his works still held their message, couched in terms and modes of thinking that are in harmony with the deepest needs and desires of modern man. Others went farther and indicated lines along which a still more profound influence of Newman could be expected in the future. The whole of the conference consequently was something of a stocktaking showing the capital realized and the promises for the future. (shrink)

We consider minimal ranks of extenders associated with Shelah cardinals by introducing witnessing numbers. Using these numbers we shall investigate effects of Shelah cardinals above themselves. MSC: 03E55.

We show that Shelah’s Eventual Categoricity Conjecture for successors follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC withLS below a strongly compact cardinalκis <κ-tame and applying the categoricity transfer of Grossberg and VanDieren [11]. These techniques also apply to measurable and weakly compact cardinals and we prove similar tameness results under those hypotheses. We isolate a dual property (...) to tameness, calledtype shortness, and show that it follows similarly from large cardinals. (shrink)

I analyze the hierarchy of large cardinals between a supercompact cardinal and an almost-huge cardinal. Many of these cardinals are defined by modifying the definition of a high-jump cardinal. A high-jump cardinal is defined as the critical point of an elementary embedding j:V→M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${j: V \to M}$$\end{document} such that M is closed under sequences of length sup{j|f:κ→κ}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sup\{{j\,|\,f: \kappa \to \kappa}\}}$$\end{document}. (...) Some of the other cardinals analyzed include the super-high-jump cardinals, almost-high-jump cardinals, Shelah-for-supercompactness cardinals, Woodin-for-supercompactness cardinals, Vopěnka cardinals, hypercompact cardinals, and enhanced supercompact cardinals. I organize these cardinals in terms of consistency strength and implicational strength. I also analyze the superstrong cardinals, which are weaker than supercompact cardinals but are related to high-jump cardinals. Among the results, I highlight the following. Vopěnka cardinals are the same as Woodin-for-supercompactness cardinals.There are no excessively hypercompact cardinals.Furthermore, I prove some results relating high-jump cardinals to forcing, as well as analyzing Laver functions for super-high-jump cardinals. (shrink)

Boykin and Jackson recently introduced a property of countable Borel equivalence relations called Borel boundedness, which they showed is closely related to the union problem for hyperfinite equivalence relations. In this paper, we introduce a family of properties of countable Borel equivalence relations which correspond to combinatorial cardinal characteristics of the continuum in the same way that Borel boundedness corresponds to the bounding number. We analyze some of the basic behavior of these properties, showing, e.g., that the property corresponding (...) to the splitting number coincides with smoothness. We then settle many of the implication relationships between the properties; these relationships turn out to be closely related to (but not the same as) the Borel Tukey ordering on cardinal characteristics. (shrink)

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)

Superstrong cardinals are never Laver indestructible. Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, 1-extendible cardinals, 0-extendible cardinals, weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, superstrongly unfoldable cardinals, Σn-reflecting cardinals, Σn-correct cardinals and Σn-extendible cardinals are never Laver indestructible. In fact, all these large cardinal properties are superdestructible: if κ exhibits any of them, with corresponding target θ, then in any forcing extension arising from nontrivial strategically <κ-closed forcing Q∈Vθ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} (...) \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Q} \in V_\theta}$$\end{document}, the cardinal κ will exhibit none of the large cardinal properties with target θ or larger. (shrink)

Journal of Mathematical Logic, Volume 22, Issue 02, August 2022. Motivated by results of Bagaria, Magidor and Väänänen, we study characterizations of large cardinal properties through reflection principles for classes of structures. More specifically, we aim to characterize notions from the lower end of the large cardinal hierarchy through the principle [math] introduced by Bagaria and Väänänen. Our results isolate a narrow interval in the large cardinal hierarchy that is bounded from below by total indescribability and from (...) above by subtleness, and contains all large cardinals that can be characterized through the validity of the principle [math] for all classes of structures defined by formulas in a fixed level of the Lévy hierarchy. Moreover, it turns out that no property that can be characterized through this principle can provably imply strong inaccessibility. The proofs of these results rely heavily on the notion of shrewd cardinals, introduced by Rathjen in a proof-theoretic context, and embedding characterizations of these cardinals that resembles Magidor’s classical characterization of supercompactness. In addition, we show that several important weak large cardinal properties, like weak inaccessibility, weak Mahloness or weak [math]-indescribability, can be canonically characterized through localized versions of the principle [math]. Finally, the techniques developed in the proofs of these characterizations also allow us to show that Hamkin’s weakly compact embedding property is equivalent to Lévy’s notion of weak [math]-indescribability. (shrink)

The large cardinal axioms of the title assert, respectively, the existence of a nontrivial elementary embedding j:Vλ→Vλ, the existence of such a j which is moreover , and the existence of such a j which extends to an elementary j:Vλ+1→Vλ+1. It is known that these axioms are preserved in passing from a ground model to a small forcing extension. In this paper the reverse directions of these preservations are proved. Also the following is shown : if V is a (...) model of ZFC and V[G] is a -generic forcing extension of V, then in V[G], V is definable using the parameter Vδ+1, where. (shrink)

This paper is a detailed study of what are traditionally called the cardinal virtues: prudence, justice, temperance and fortitude. I defend what I call the Cardinality Thesis, that the traditional four and no others are cardinal. I define cardinality in terms of three sub-theses, the first being that the cardinal virtues are jointly necessary for the possession of every other virtue, the second that each of the other virtues is a species of one of the four (...) cardinals, and the third that many of the other virtues are also auxiliaries of one or more cardinals. I provide abstract arguments for each sub-thesis, followed by illustration from concrete cases. I then use these results to shed light on the two fundamental problems of the acquisition of the virtues and their unity, proving some further theses in the latter case. (shrink)

We give some results concerning various generalized continuum cardinals. The results answer some natural questions which have arisen in preparing a new edition of 5. To make the paper self-contained we define all of the cardinal functions that enter into the theorems here. There are many problems concerning these new functions, and we formulate some of the more important ones.

We prove that each proper ideal in the lattice of axiomatic, resp. standard strengthenings of the intuitionistic propositional logic is of cardinality 20. But, each proper ideal in the lattice of structural strengthenings of the intuitionistic propositional logic is of cardinality 220. As a corollary we have that each of these three lattices has no atoms.

We present equiconsistency results at the level of subcompact cardinals. Assuming SBHδ, a special case of the Strategic Branches Hypothesis, we prove that if δ is a Woodin cardinal and both □ and □δ fail, then δ is subcompact in a class inner model. If in addition □ fails, we prove that δ is Π12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi_1^2}$$\end{document} subcompact in a class inner model. These results are optimal, and lead to equiconsistencies. As a (...) corollary we also see that assuming the existence of a Woodin cardinal δ so that SBHδ holds, the Proper Forcing Axiom implies the existence of a class inner model with a Π12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi_1^2}$$\end{document} subcompact cardinal. Our methods generalize to higher levels of the large cardinal hierarchy, that involve long extenders, and large cardinal axioms up to δ is δ+ supercompact for all n < ω. We state some results at this level, and indicate how they are proved. (shrink)

Let κ denote a regular uncountable cardinal and NS the normal ideal of nonstationary subsets of κ. Our results concern the well-known open question whether NS fails to be κ + -saturated, i.e., are there κ + stationary subsets of κ with pairwise intersections nonstationary? Our first observation is: Theorem. NS is κ + -saturated iff for every normal ideal J on κ there is a stationary set $A \subseteq \kappa$ such that $J = NS \mid A = \{X (...) \subseteq \kappa:X \cap A \in NS\}$ . Turning our attention to large cardinals, we extend the usual (weak) Mahlo hierarchy to define "greatly Mahlo" cardinals and obtain the following: Theorem. If κ is greatly Mahlo then NS is not κ + -saturated. Theorem. If κ is ordinal Π 1 1 -indescribable (e.g., weakly compact), ethereal (e.g., subtle), or carries a κ-saturated ideal, then κ is greatly Mahlo. Moreover, there is a stationary set of greatly Mahlo cardinals below any ordinal Π 1 1 -indescribable cardinal. These methods apply to other normal ideals as well; e.g., the subtle ideal on an ineffable cardinal κ is not κ + -saturated. (shrink)

There has recently been work by multiple groups in extracting the properties associated with cardinal invariants of the continuum and translating these properties into similar analogous combinatorial properties of computational oracles. Each property yields a highness notion in the Turing degrees. In this paper we study the highness notions that result from the translation of the evasion number and its dual, the prediction number, as well as two versions of the rearrangement number. When translated appropriately, these yield four new (...) highness notions. We will define these new notions, show some of their basic properties and place them in the computability-theoretic version of Cichoń's diagram. (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)

We study the values of the higher dimensional cardinal characteristics for sets of functions $f:\omega ^\omega \to \omega ^\omega $ introduced by the second author in [8]. We prove that while the bounding numbers for these cardinals can be strictly less than the continuum, the dominating numbers cannot. We compute the bounding numbers for the higher dimensional relations in many well known models of $\neg \mathsf {CH}$ such as the Cohen, random and Sacks models and, as a byproduct show (...) that, with one exception, for the bounding numbers there are no $\mathsf {ZFC}$ relations between them beyond those in the higher dimensional Cichoń diagram. In the case of the dominating numbers we show that in fact they collapse in the sense that modding out by the ideal does not change their values. Moreover, they are closely related to the dominating numbers $\mathfrak {d}^\lambda _\kappa $. (shrink)

We prove several theorems about the cardinal $\mathfrak{g}$ associated with groupwise density. With respect to a natural ordering of families of nond-ecreasing maps fromω toω, all families of size $< \mathfrak{g}$ are below all unbounded families. With respect to a natural ordering of filters onω, all filters generated by $< \mathfrak{g}$ sets are below all non-feeble filters. If $\mathfrak{u}< \mathfrak{g}$ then $\mathfrak{b}< \mathfrak{u}$ and $\mathfrak{g} = \mathfrak{d} = \mathfrak{c}$ . (The definitions of these cardinals are recalled in the introduction.) (...) Finally, some consequences deduced from $\mathfrak{u}< \mathfrak{g}$ by Laflamme are shown to be equivalent to $\mathfrak{u}< \mathfrak{g}$. (shrink)

Local sentences were introduced by Ressayre in [6] who proved certain remarkable stretching theorems establishing the equivalence between the existence of finite models for these sentences and the existence of some infinite well ordered models. Two of these stretching theorems were only proved under certain large cardinal axioms but the question of their exact strength was left open in [4]. Here we solve this problem, using a combinatorial result of J. H. Schmerl [7]. In fact, we show that the (...) stretching principles are equivalent to the existence of n -Mahlo cardinals for appropriate integers n. This is done by proving first that for every integer n, there is a local sentence φn having well ordered models of order type τ, for every infinite ordinal τ > ω which is not an n -Mahlo cardinal. (shrink)

Nietzsche’s cardinal ideas - God is Dead, Übermensch and Eternal Return of the Same - are approached here from the perspective of psychiatric phenomenology rather than that of philosophy. A revised diagnosis of the philosopher’s mental illness as manic-depressive psychosis forms the premise for discussion. Nietzsche conceived the above thoughts in close proximity to his first manic psychotic episode, in the summer of 1881, while staying in Sils-Maria (Swiss Alps). It was the anniversary of his father’s death, and also (...) of the break-up of his friendship with Wagner, the most important relationship in his life. Despite having been acquainted with these ideas from reading philosophy and literature, Nietzsche created them de novo and imbued them with very personal meaning. Surprisingly, he never defined or explained his cardinal thoughts in his published writings, perhaps because rationally he could not. A resultant hermeneutic vacuum provoked an avalanche of interpretations in secondary literature. But could these ideas be delusions? A current definition of delusion is challenged, and an attempt is made at a limited comparison between delusion, scientific/philosophical doctrine and poetic creation. It is also argued that psychosis is a way of re-living trauma, and delusions can therefore be seen as a form of reasoning that helps to make sense of the world in a state of psychotic disintegration. Far from being false beliefs, delusions are a true expression of one’s innermost feelings and pain, albeit indirectly. The relationship between early parental loss and repeated trauma, psychosis and creativity is also explored. Indo-Pacific Journal of Phenomenology , Volume 8, Edition 1 May 2008. (shrink)

Assuming AD + DC, we characterize the self-dual boldface pointclasses which are strictly larger than the pointclasses contained in them: these are exactly the clopen sets, the collections of all sets of Wadge rank [Formula: see text], and those of Wadge rank [Formula: see text] when ξ is limit.

We summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants. Our Main Theorem shows how to apply these methods to many cardinals simultaneously and exactly control which cardinals are supercompact and which are only strongly compact in a forcing extension. Depending upon the method, the surviving non-supercompact strongly compact cardinals can be strong cardinals, have trivial Mitchell rank or even contain a club disjoint from the set of measurable cardinals. These results improve (...) and unify Theorems 1 and 2 of [5], due to the first author. (shrink)