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)

Given a Mahlo cardinal κ and a regular ε such that $\omega_1 we show that $\diamond_\kappa (cf = \epsilon)$ holds in V provided that there are only non-stationarily many $\beta , with o(β) ≥ ε in K.

We show that a cardinal κ is a (strongly) Mahlo cardinal if and only if there exists a nontrivial κ-complete κ-normal ideal on κ. Also we show that if κ is Mahlo and λ ≥ κ and $\lambda^{ then there is a nontrivial κ-complete κ-normal fine ideal on P κ (λ). If κ is the successor of a cardinal, we consider weak κ-normality and prove that if κ = μ + and μ is a regular cardinal then (1) (...) $\mu^{ if and only if there is a nontrivial κ-complete weakly κ-normal ideal on κ, and (2) if $\mu^{ then there is a nontrivial κ-complete weakly κ-normal fine ideal on P κ (λ). (shrink)

This paper introduces the theme of killing‐them‐softly between set‐theoretic universes. The main theorems show how to force to reduce the large cardinal strength of a cardinal to a specified desired degree. The killing‐them‐softly theme is about both forcing and the gradations in large cardinal strength. Thus, I also develop meta‐ordinal extensions of the hyper‐inaccessible and hyper‐Mahlo degrees. This paper extends the work of Mahlo to create new large cardinals and also follows the larger theme of exploring interactions (...) between large cardinals and forcing central to modern set theory. (shrink)

If F is a normal filter on a regular uncountable cardinal κ, let |f| be the F-norm of an ordinal function f. We introduce the class of positive ordinal operations and prove that if F is a positive operation then |F(f)| ≥ F(|f|). For each $\eta let f η be the canonical ηth function. We show that if F is a Σ operation then F(f η ) = f F(η) . As an application we show that if κ is greatly (...) Mahlo then there are normal filters on κ of order greater than κ +. (shrink)

This paper will define a new cardinal called aStationary Cardinal. We will show that every weakly∏ 1 1 -indescribable cardinal is a stationary cardinal, every stationary cardinal is a greatly Mahlo cardinal and every stationary set of a stationary cardinal reflects. On the other hand, the existence of such a cardinal is independent of that of a∏ 1 1 -indescribable cardinal and the existence of a cardinal such that every stationary set reflects is also independent of that of a (...) stationary cardinal. As applications, we will show thatV=L implies ◊ κ 1 holds if κ is∏ 1 1 -indescribable and so forth. (shrink)

The generic Vopěnka principle, we prove, is relatively consistent with the ordinals being non-Mahlo. Similarly, the generic Vopěnka scheme is relatively consistent with the ordinals being definably non-Mahlo. Indeed, the generic Vopěnka scheme is relatively consistent with the existence of a \-definable class containing no regular cardinals. In such a model, there can be no \-reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, Gitman and Schindler.

Subtle cardinals were first introduced in a paper by Jensen and Kunen [JK]. They show that ifκis subtle then ◇κholds. Subtle cardinals also play an important role in [B1], where Baumgartner proposed that certain large cardinal properties should be considered as properties of their associated normal ideals. He shows that in the case of ineffables, the ideals are particularly useful, as can be seen by the following theorem,κis ineffable if and only ifκis subtle andΠ½-indescribableandthe subtle andΠ½-indescribable ideals cohere, (...) i.e. they generate a proper, normal ideal.In this paper we examine properties of subtle cardinals and consider methods of forcing that destroy the property of subtlety while maintaining other properties. The following is a list of results.1) We relativize the following two facts about subtle cardinals:i) ifκisn-subtle then {α<κ:αis notn-subtle} isn-subtle, andii) ifκis -subtle then {α<κ:αisn-subtle} is in the -subtle filter to subsets ofκ:i′) ifAis ann-subtle subset ofκthen {α ϵ A:A∩αis notn-subtle} isn-subtle, andii′) ifAis an -subtle subset ofκthen {α ϵ A:A∩αisn-subtle} is -subtle.2) We show that although a stationary limit of subtles is subtle, a subtle limit of subtles is not necessarily 2-subtle.3) In §3 we use the technique of forcing to turn a subtle cardinal into aκ-Mahlo cardinal that is no longer subtle.4) In §4 we extend the results of §3 by showing how to turn an -subtle cardinal into ann-subtle cardinal that is no longer -subtle. (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)

We consider a weak version of Schindler’s remarkable cardinals that may fail to be ${{\rm{\Sigma }}_2}$-reflecting. We show that the ${{\rm{\Sigma }}_2}$-reflecting weakly remarkable cardinals are exactly the remarkable cardinals, and that the existence of a non-${{\rm{\Sigma }}_2}$-reflecting weakly remarkable cardinal has higher consistency strength: it is equiconsistent with the existence of an ω-Erdős cardinal. We give an application involving gVP, the generic Vopěnka principle defined by Bagaria, Gitman, and Schindler. Namely, we show that gVP + “Ord (...) is not ${{\rm{\Delta }}_2}$-Mahlo” and ${\text{gVP}}$ + “there is no proper class of remarkable cardinals” are both equiconsistent with the existence of a proper class of ω-Erdős cardinals, extending results of Bagaria, Gitman, Hamkins, and Schindler. (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)

Assuming the existence of a Mahlo cardinal, we produce a generic extension of Gödel’s constructible universe L, in which the \ holds and the transfer principles \ \rightarrow \) and \ \rightarrow \) fail simultaneously. The result answers a question of Silver from 1971. We also extend our result to higher gaps.

The following problem is studied: How large and how small can the Löwenheim and Hanf numbers of unbounded logics be in relation to the most common large cardinals? The main result is that the Löwenheim number of the logic with the Härtig-quantifier can be consistently put in between any two of the first weakly inaccessible, the first weakly Mahlo, the first weakly compact, the first Ramsey, the first measurable and the first supercompact cardinals.

An outline is given of the proof that the consistency of a κ⁺-Mahlo cardinal implies that of the statement that I[ω₂] does not include any stationary subsets of Cof(ω₁). An additional discussion of the techniques of this proof includes their use to obtain a model with no ω₂-Aronszajn tree and to add an ω₂-Souslin tree with finite conditions.

La dotta ignoranza -- Le congetture -- Il Dio nascosto -- La ricerca di Dio -- La filiazione di Dio -- Il dono del Padre dei lumi -- Congettura sugli ultimi giorni -- Dialogo sulla Genesi -- Difesa della dotta ignoranza -- La sapienza -- La mente -- Gli esperimenti con la bilancia -- La visione di Dio -- Il berillo -- L'ugaglianza -- Il principio -- Il potere che è -- il non-altro -- La caccia della sapienza -- Il (...) gioco della palla -- Compendio -- Il vertice della contemplazione -- La quadratura del cerchio -- I complementi matematici -- La perfezione matematica. (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)

Cardinal Mercier’s Manual of Modern Scholastic Philosophy is a standard work, prepared at the Higher Institute of Philosophy, Louvain, mainly for the use of clerical students in Catholic Seminaries. Though undoubtedly elementary, it contains a clear, simple, and methodological exposition of the principles and problems of every department of philosophy, and its appeal is not to any particular class, but broadly human and universal. Volume I includes a general introduction to philosophy and sections on cosmology, psychology, criteriology, and metaphysics or (...) ontology. (shrink)

Thus emerges the paradox of All Israel's destiny. Some of the children of Israel gathered in a state like all others—no more and no less and this is legitimate and necessary. This state was founded by the children of the People whom God called not to be like the others, but, rather for the others, because of His design for universal salvation. What is true for the people who have settled in this state which was recreated for the Jews, is (...) true for the members, of the Jewish people who are dispersed throughout the Diaspora as citizens of various countries. But this vocation to testify is in the hands of God, and not in the power of man. The destiny of Israel is that its destiny overtakes it over and over. (shrink)

This paper concerns late 1920 s attempts to construct unitary theories of gravity and electromagnetism. A first attempt using a non-standard connection—with torsion and zero-curvature—was carried out by Albert Einstein in a number of publications that appeared between 1928 and 1931. In 1929, Tullio Levi-Civita discussed Einstein’s geometric structure and deduced a new system of differential equations in a Riemannian manifold endowed with what is nowadays known as Levi-Civita connection. He attained an important result: Maxwell’s electromagnetic equations and the gravitational (...) equations were obtained exactly, while Einstein had deduced them only as a first order approximation. A main feature of Levi-Civita’s theory is the essential use of the Ricci’s rotation coefficients, introduced by Gregorio Ricci Curbastro many years before. We trace the history of Ricci’s coefficients that are still used today, and highlight their geometric and mechanical meaning. (shrink)

"Individuum und Kosmos in der Philosophie der Renaissance" (1927) schreibt ein Stück philosophischer Problemgeschichte und geht der Frage nach, "ob und inwiefern die Gedankenbewegung des 15. und 16. Jahrhunderts bei aller Mannigfaltigkeit der Problemansätze und bei aller Divergenzen der Lösungen eine in sich geschlossene Einheit bildet". Provoziert durch Burckhardts Renaissancestudie, die die Philosophie der Zeit unberücksichtigt läßt, versucht Cassirer nachzuweisen, daß auch die Renaissancephilosophie Teil einer "geistigen Gesamtbewegung" ist und eigene systematische Mittelpunkte besitzt.

Rousseau’s political project consists in ensuring that the citizens of the social contract, in uniting with each other, preserve their ability to self-legislate, or be autonomous. For this to work, however, members of the social contract would need to feel intrinsically linked to the political whole. This essay investigates what that feeling might be and how it can be grown. I argue that Rousseau develops a model of the energy or character of the being capable of autonomy, capable of experiencing (...) themselves as part of the whole. That energy is a pathos of vigour, a strong sentiment and way of being that I develop from Rousseau’s educational precepts in Émile, which makes the citizen feel free and robust in dependence and boundedness. Autonomy, then, comes from the active exercise of oneself, physically and mentally, in an environment bounded by things, and this results in a sentiment of vigour, a vitality that produces confidence and poise, which then entices further activity. (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)

Cardinal Mercier’s Manual of Modern Scholastic Philosophy is a standard work, prepared at the Higher Institute of Philosophy, Louvain, mainly for the use of clerical students in Catholic Seminaries. Though undoubtedly elementary, it contains a clear, simple, and methodological exposition of the principles and problems of every department of philosophy, and its appeal is not to any particular class, but broadly human and universal. Volume II contains sections on natural theology, logic, ethics and outlines of the history of philosophy.

Cardinal Mercier's A Manual of Modern Scholastic Philosophy is a standard work, prepared at the Higher Institute of Philosophy, Louvain, mainly for the use of clerical students in Catholic seminaries. Though undoubtedly elementary, it contains a clear, simple, and methodological exposition of the principles and problems of every department of philosophy, and its appeal is not to any particular class, but broadly human and universal. Volume I includes a general introduction to philosophy and sections on cosmology, psychology, criteriology, and metaphysics (...) or ontology. Volume II contains sections on natural theology, logic, ethics, and outlines of the history of philosophy. (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)