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)

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 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)

Let k be an infinite cardinal. A subset of $(^k k)^n $ is a $\Sigma _1^1 $ -subset if it is the projection p[T] of all cofinal branches through a subtree T of $(lt;kk)^{n + 1} $ of height k. We define $\Sigma _k^1 - ,\Pi _k^1 $ - and $\Delta _k^1$ subsets of $(^k k)^n $ as usual. Given an uncountable regular cardinal k with k = k (...) show that there is a 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)

Philippe Roy-Lysencourt | : Dans cet article, Philippe Roy-Lysencourt présente les rapports personnels entre Pie X et son Secrétaire d’État, le cardinal Rafael Merry del Val. Il considère tout d’abord la période qui va de la première rencontre entre les deux hommes jusqu’à la nomination de Merry del Val comme Secrétaire d’État ; il examine ensuite les relations de travail et les relations personnelles entre les deux hommes, avant de relater leur dernière rencontre, au chevet de Pie (...) X, qui illustre d’une façon particulière la profondeur de leur amitié. À partir de sources inédites, Philippe Roy-Lysencourt montre que les relations entre les deux hommes étaient fondées sur une estime, un respect et une admiration réciproques, qu’elles étaient caractérisées par une grande confiance mutuelle et que, finalement, les rapports entre les deux hommes étaient si étroits que l’histoire et l’historien ne peuvent dissocier les noms de Pie X et du cardinal Merry del Val. | : In this article, Philippe Roy-Lysencourt presents the personal relationships between Pius X and his Secretary of State, Cardinal Rafael Merry del Val. He first takes into consideration the period that goes from the first encounter between the two men until Merry del Val’s appointment as Secretary of State ; he then examines the working and personal relations between the two men, before discussing their last encounter, at Pius X’s bedside, which illustrates in a particular way the depth of their friendship. Working from unpublished sources, Philippe Roy-Lysencourt shows that the relationship between the two men was based on a reciprocal esteem, respect and admiration, that it was characterized by great mutual trust, and that finally this relationship was so close that history and the historian cannot dissociate the names of Pius X and Cardinal Merry del Val. (shrink)

We study closure properties of measurable ultrapowers with respect to Hamkin's notion of freshness and show that the extent of these properties highly depends on the combinatorial properties of the underlying model of set theory. In one direction, a result of Sakai shows that, by collapsing a strongly compact cardinal to become the double successor of a measurable cardinal, it is possible to obtain a model of set theory in which such ultrapowers possess the strongest possible closure properties. (...) In the other direction, we use various square principles to show that measurable ultrapowers of canonical inner models only possess the minimal amount of closure properties. In addition, the techniques developed in the proofs of these results also allow us to derive statements about the consistency strength of the existence of measurable ultrapowers with non-minimal closure properties. (shrink)

Given an uncountable regular cardinal κ, we study the structural properties of the class of all sets of functions from κ to κ that are definable over the structure 〈H,∈〉 by a Σ1-formula with parameters. It is well known that many important statements about these classes are not decided by the axioms of ZFC together with large cardinal axioms. In this paper, we present other canonical extensions of ZFC that provide a strong structure theory for these classes. These (...) axioms are variations of the Maximality Principle introduced by Stavi and Väänänen and later rediscovered by Hamkins. (shrink)

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\fancyscript{L}}$$\end{document} be a finite first-order language and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\langle{\fancyscript{M}_n} \,|\, {n < \omega}\rangle}$$\end{document} be a sequence of finite \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\fancyscript{L}}$$\end{document}-models containing models of arbitrarily large finite cardinality. If the intersection of less than continuum-many dense open subsets of Cantor Space ω2 is non-empty, then there is a non-principal ultrafilter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} (...) \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\fancyscript{U}}$$\end{document} over ω such that the corresponding ultraproduct \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\prod_\fancyscript{U}\fancyscript{M}_n}$$\end{document} has an automorphism that is not induced by an element of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\prod_{n<\omega}{\rm Aut}(\fancyscript{M}_n)}$$\end{document}. (shrink)

The sense of shared values is a specific aspect of human sociality. It originates from reciprocal social exchanges that include imitation, and empathy, but also negotiation from which meanings, values and norms are eventually constructed with others. Research suggests that this process starts from birth via imitation and mirroring processes that are important foundations of sociality providing a basic sense of social connectedness and mutual acknowledgment with others. From the second month, mirroring, imitative and other contagious responses are bypassed. Neonatal (...) imitation gives way to first signs of reciprocation (primary intersubjectivity), and joint attention in reference to objects (secondary intersubjectivity). We review this development and propose a third level of intersubjectivity, that is the emergence of values that are jointly represented and negotiated with others, as well as the development of an ethical stance accompanying emerging theories of mind from about 4 years of age. We propose that tertiary intersubjectivity is an ontogenetically new process of value negotiation and mutual recognition that are the cardinal trademarks of human sociality. (shrink)

Recently, Gitik, Kanovei and the first author proved that for a classical Prikry forcing extension the family of the intermediate models can be parametrized by $\mathscr{P}(\omega)/\mathrm{finite}$. By modifying the standard Prikry tree forcing we define a Prikry-type forcing which also singularizes a measurable cardinal but which is minimal, i.e., there are \emph{no} intermediate models properly between the ground model and the generic extension. The proof relies on combining the rigidity of the tree structure with indiscernibility arguments resulting from the (...) normality of the associated measures. (shrink)

We introduce and study some variants of a notion of canonical set theoretical truth. By this, we mean truth in a transitive proper class model M of ZFC that is uniquely characterized by some $$\in$$ ∈ -formula. We show that there are interesting statements that hold in all such models, but do not follow from ZFC, such as the ground model axiom and the nonexistence of measurable cardinals. We also study a related concept in which we only require M to (...) be fixed up to elementary equivalence. We show that this theory-canonicity also goes beyond provability in ZFC, but it does not rule out measurable cardinals and it does not fix the size of the continuum. (shrink)

Le Conseiller d'Etat ou Recueil général de la politique moderne est l'un des grands traités politiques du XVIIe siècle. Son auteur, Philippe de Béthune, diplomate réputé et fin connaisseur de la vie publique, a écrit un livre étonnant de philosophie politique, de stratégie, d'économie et d'urbanisme. Œuvre d 'un homme de cour, habitué à côtoyer les princes, il est le produit de la réflexion et de l'expérience d'un personnage singulier, mais aussi d'une époque insolite de mutation entre deux mondes, (...) - celui de la Renaissance, des guerres de religion et des troubles de la Ligue, et celui de l'absolutisme monarchique et de l'État moderne, - marquée par une transformation profonde des valeurs politiques et morales. Personnalité peu commune, qui a voyagé à travers toute l'Europe, qui par deux fois a été ambassadeur à Rome et qui a bénéficié d'une longévité remarquable, Philippe de Béthune est le frère puîné de Sully Il naît sous Charles IX, commence sa carrière comme mignon, dans l'entourage de Joyeuse, l'archi-favori d'Henri III, sert Henri lV Louis XIII, Richelieu et termine sa vie sous le ministériat de Mazarin. Publié en 1633, avec l'accord du cardinal ministre, Le Conseiller d'Etat n 'a jamais été réédité depuis 1684. Cette nouvelle édition est précédée d'une longue introduction sur l'art politique de la première moitié du XVlle siècle et le paysage mental de l'époque. (shrink)

We prove a number of results on the determinacy of σ-projective sets of reals, i.e., those belonging to the smallest pointclass containing the open sets and closed under complements, countable unions, and projections. We first prove the equivalence between σ-projective determinacy and the determinacy of certain classes of games of variable length <ω^2 (Theorem 2.4). We then give an elementary proof of the determinacy of σ-projective sets from optimal large-cardinal hypotheses (Theorem 4.4). Finally, we show how to generalize the (...) proof to obtain proofs of the determinacy of σ-projective games of a given countable length and of games with payoff in the smallest σ-algebra containing the projective sets, from corresponding assumptions (Theorem 5.1, Theorem 5.4). (shrink)

Imitation and mirroring processes are necessary but not sufficient conditions for children to develop human sociality. Human sociality entails more than the equivalence and connectedness of perceptual experiences. It corresponds to the sense of a shared world made of shared values. It originates from complex ‘open’ systems of reciprocation and negotiation, not just imitation and mirroring processes that are by definition ‘closed’ systems. From this premise, we argue that if imitation and mirror processes are important foundations for sociality, human inter-subjectivity (...) develops primarily in reciprocation, not just imitation. Imitation provides a basic sense of social connectedness and mutual acknowledgment of existing with others that are ‘like me.’ However, it does not allow for the co-construction of meanings with others. For human sociality to develop, imitation and mirroring processes need to be supplemented by an open system of reciprocation. Developmental research shows that from the second month, mirroring, imitative, and other contagious emotional responses are by-passed. Imitation gives way to first signs of reciprocation (primary intersubjectivity), joint attention to objects (secondary intersubjectivity), the emergence of values that are jointly represented and negotiated with others (tertiary intersubjectivity), and eventually the development of an ethical stance accompanying theories of mind by 4 years of age. We review this development and propose that if mirroring processes enable individuals to bridge their subjective experiences, human inter-subjectivity proper develops from reciprocal social exchanges that lead to value negotiation and mutual recognition, both cardinal trademarks of human sociality. (shrink)

We investigate a variety of cut and choose games, their relationship with (generic) large cardinals, and show that they can be used to characterize a number of properties of ideals and of partial orders: certain notions of distributivity, strategic closure, and precipitousness.

We contribute to the study of generalizations of the Perfect Set Property and the Baire Property to subsets of spaces of higher cardinalities, like the power set ${\mathcal {P}}({\lambda })$ of a singular cardinal $\lambda $ of countable cofinality or products $\prod _{i<\omega }\lambda _i$ for a strictly increasing sequence $\langle {\lambda _i}~\vert ~{i<\omega }\rangle $ of cardinals. We consider the question under which large cardinal hypothesis classes of definable subsets of these spaces possess such regularity properties, focusing (...) on rank-into-rank axioms and classes of sets definable by $\Sigma _1$ -formulas with parameters from various collections of sets. We prove that $\omega $ -many measurable cardinals, while sufficient to prove the perfect set property of all $\Sigma _1$ -definable sets with parameters in $V_\lambda \cup \{V_\lambda \}$, are not enough to prove it if there is a cofinal sequence in $\lambda $ in the parameters. For this conclusion, the existence of an I2-embedding is enough, but there are parameters in $V_{\lambda +1}$ for which I2 is still not enough. The situation is similar for the Baire property: under I2 all sets that are $\Sigma _1$ -definable using elements of $V_\lambda $ and a cofinal sequence as parameters have the Baire property, but I2 is not enough for some parameter in $V_{\lambda +1}$. Finally, the existence of an I0-embedding implies that all sets that are $\Sigma ^1_n$ -definable with parameters in $V_{\lambda +1}$ have the Baire property. (shrink)

We investigate the notion of strong measure zero sets in the context of the higher Cantor space $2^\kappa $ for $\kappa $ at least inaccessible. Using an iteration of perfect tree forcings, we give two proofs of the relative consistency of $$\begin{align*}|2^\kappa| = \kappa^{++} + \forall X \subseteq 2^\kappa:\ X \textrm{ is strong measure zero if and only if } |X| \leq \kappa^+. \end{align*}$$ Furthermore, we also investigate the stronger notion of stationary strong measure zero and show that the equivalence (...) of the two notions is undecidable in ZFC. (shrink)

In topological spaces, the relation of extended contact is a ternary relation that holds between regular closed subsets A, B and D if the intersection of A and B is included in D. The algebraic counterpart of this mereotopological relation is the notion of extended contact algebra which is a Boolean algebra extended with a ternary relation. In this paper, we are interested in the relational representation theory for extended contact algebras. In this respect, we study the correspondences between point-free (...) and point-based models of space in terms of extended contact. More precisely, we prove new representation theorems for extended contact algebras. (shrink)

Straightening the current ‘value-laden turn’ (VLT) in the philosophical literature on values in science, and reviving the legacy of the value-free ideal of science (VFI), this paper argues that the influence of extra-scientific values should be minimised—not excluded—in the core phase of scientific inquiry where claims are accepted or rejected. Noting that the original arguments for the VFI (ensuring the truth of scientific knowledge, respecting the autonomy of science results users, preserving public trust in science) have not been satisfactorily addressed (...) by proponents of the VLT, it proposes four prerequisites which any model for values in the acceptance/rejection phase of scientific inquiry should respect, coming from the fundamental requirement to distinguish between facts and values: (1) the truth of scientific knowledge must be ensured; (2) the uncertainties associated with scientific claims must be stated clearly; (3) claims accepted into the scientific corpus must be distinguished from claims taken as a basis for action. An additional prerequisite of (4) simplicity and systematicity is desirable, if the model is to be applicable. Methodological documents from international institutions and regulation agencies are used to illustrate the prerequisites. A model combining Betz’s conception (stating uncertainties associated with scientific claims) and Hansson’s corpus model (ensuring the truth of the scientific corpus and distinguishing it from other claims taken as a basis for action) is proposed. Additional prerequisites are finally suggested for future research, stemming from the requirement for philosophy of science to self-reflect on its own values: (5) any model for values in science must be descriptively and normatively relevant; and (6) its consequences must be thoroughly assessed. (shrink)

The seemingly "inextricable" mutual entanglement of history and the play of power dissolves itself when thinking, under the sway of technique, moves beyond the horizon of metaphysic, and turns to question the unconcealment of being, that is, when man lets time become the law of his essence. Only so would a history, i.e. freedom, happen again, and rescue modern man from his otherwise destinyless age.

This article presents the Arabic text of al-Q's short geometric treatise, The Book of the centres of the tangent circles on lines, by the method of analysis, with translation and a mathematical commentary. In this treatise al-Q solves by analysis an ordered set of eight problems where the goal is to locate, on a given line, the centre of a circle which is tangent to two given elements, which may be points, straight lines or circles. For example, in the first (...) problem he wanted to find the centre of a circle which contains two given points. In fact, al-Q establishes the locus of the solutions, and this locus can be a straight line or a conic. The treatise is reminiscent of one of the purposes of al-Q's work on the perfect compass, namely to assign to the conics the status of regular lines. (shrink)

Animal research suggests that central serotonergic neurons are involved in behavioral suppression, particularly anxiety-related inhibition. The hypothesis linking decreased serotonin transmission to reduced anxiety as the mechanism in the anxiolytic activity of benzodiazepines conflicts with most clinical observations. Serotonin antagonists show no marked capacity to alleviate anxiety. On the other hand, clinical signs of reduced serotonergic transmission (low 5-HIAA levels in the cerebrospinal fluid) are frequently associated with aggressiveness, suicide attempts, and increased anxiety. The target article attempts to reconcile such (...) human and animal findings by investigating whether anxiety reduction or increased impulsivity is more Likely to account for animal behavioral changes associated with decreased serotonergic transmission. The effects of manipulating central serotonin in experimental anxiety paradigms in animals (punishment, extinction, novelty) are reviewed and compared with the effects of antianxiety drugs. Anxiety seems neither necessary nor sufficient to induce control by serotonergic neurons on behavior. Further evidence suggests that behavioral effects of anxiolytics thought to be mediated by decreases in anxiety are not caused by the ability of these drugs to reduce serotonin transmission. Blockade of serotonin transmission, especially at the level of the substantia nigra, results in a shift of behavior toward facilitation of responding. This behavioral shift is particularly marked when there is competition between acting and restraining response tendencies and when obstacles prevent the immediate attainment of an anticipated reward. It is proposed that serotonergic neurons are involved not only in behavioral arousal but also in enabling the organism to arrange or tolerate delay before acting. Decreases in serotonin transmission seem to be associated with the increased performance of behaviors that are usually suppressed, though not necessarily because of the alleviation of anxiety, which might contribute to the suppression. (shrink)

Il est encore paradoxal de reconnaître aux livres des Chroniques le statut d'œuvre littéraire. Longtemps considéré comme de « piètre fiabilité » par rapport au récit parallèle des livres de Samuel et des Rois, ces livres apparaissent aussi sans originalité littéraire par rapport notamment à l'art consommé des récits des livres de Samuel. Dans le sillage de l'Art du récit biblique, de Robert Alter, Ph. Abadie tente de faire ressortir la richesse et la variété des procédés d'écriture qui font du (...) livre des Chroniques une œuvre littéraire véritable. Celle-ci tient fondamentalement à un projet global que révèlent et confirment des « unités kérygmatiques » structurantes et l'ordonnancement de grands ensembles selon divers procédés : technique de répétition et système de périodisation chronologique notamment.It is still paradoxical to recognize the statute of literary work for the books of Chronicles. Long time considered as of “poor reliability” in comparison with the parallel history of the books of Samuel and Kings, these books also seem to be without literary originality in comparison with, notably, the consummate art of the narratives in the books of Samuel. In the wake of “The Art of Biblical Narrative” by Robert Alter, Ph. Abadie tries to bring out the richness and the variety of the writing procedures that make the book of Chronicles a truly literary work. It fundamentally holds to a global project that reveals and confirms some “kerygmatic unities,” structuring and ordering great ensembles according to diverse procedures: especially a technique of repetition and system of chronological periodisation. (shrink)

RésuméDans son traité intituléLe Planisphère, Ptolémée présente une méthode pour représenter une sphère sur un plan, selon des principes compatibles avec ce qu'on nomme aujourd'hui la projection stéréographique. Mais cette dernière ne sera traitée mathématiquement, en tant que telle, que bien plus tard, au IXẹsiècle, dans l'œuvre d'al-Farghānī qui démontrera notamment la propriété fondamentale de cette projection. Ce n'est qu'au Xesiècle qu'al-Qūhī et Ibn Sahl écriront une première théorie générale des projections de la sphère. Cet article analyse les raisons qui (...) permettent d'affirmer que Ptolémée n'a pas conçu mathématiquement de projection, et que cette conception n'a pu être possible qu'avec l'aide de la définition de la surface conique telle qu'Apollonius l'a exposée dans sesConiques. (shrink)

Des premiers rites baptismaux à la confession moderne, les références au christianisme sont constantes dans l'œuvre de Michel Foucault. Cette constance s'inscrit dans un questionnement philosophique plus large sur notre actualité : comprendre le rapport que nous avons aujourd'hui à nous-mêmes demande de s'interroger sur les actes de vérité que l'Occident a instaurés depuis les premiers siècles chrétiens. Que faut-il dire et manifester de soi pour être transformé dans son être, pardonné, sauvé, jugé ou guéri)? Ce livre propose une étude (...) critique de l'ensemble des lectures chrétiennes de Foucault, avec une attention particulière portée au cours Du gouvernement des vivants. Ni chronologique ni thématique, le parcours suivi espère retrouver la logique d'un travail à la fois philosophique et historique : quand et comment le christianisme a-t-il été constitué par Foucault en objet de recherche, avec quelles pratiques de lecture et quelles conséquences sur l'interprétation)? (shrink)