In lieu of an abstract, here is a brief excerpt of the content:Message to Buddhists for the Feast of Vesakh 2007:Christians and Buddhists: Educating Communities to Live in Harmony and PeacePaul Cardinal Poupard, President and Archbishop Pier Luigi Celata, SecretaryDear Buddhist Friends,1. On the occasion of the festival of Vesakh, I am writing to Buddhist communities in different parts of the world to convey my own good wishes, as well as those of the Pontifical Council for Interreligious Dialogue.2. (...) We, Catholics and Buddhists, enjoy a good relationship and our contacts, collaboration and implementation of diverse programmes have helped to deepen our understanding of each other. Dialogue is the sure path to fruitful interreligious relations. It deepens respect and nurtures the desire to live in harmony with others.3. The Second Vatican Council teaches that the entire human race shares a common origin and a common destiny: God, our Creator and the goal of our earthly pilgrimage. Similarly, Pope Benedict XVI, in his 2007 Message for the World Day of Peace, observed: "As one created in the image of God, each individual human being has the dignity of a person; he or she is not just something, but someone, capable of self-knowledge, self-possession, free self-giving and entering into communion with others" (n. 2).4. Building a community requires concrete gestures which reflect the respect for the dignity of others. Furthermore, as religious people, we are convinced that "there is a moral logic which is built into human life and which makes possible dialogue between individuals and peoples" (ibid, n. 3). Yet, there are people today who still need to learn about others and other people's beliefs in order to overcome prejudices and misunderstandings. This sad reality, if it is to be overcome, demands much effort [End Page 131] on the part of both civic and religious leaders. Even in places where people experience daily the ravages of war, fueled by sentiments of hatred and vengeance, trust can be restored. Together we can help to create the space and the opportunities for people to talk, listen, share regrets and offer forgiveness for each other's past mistakes.5. Education for peace is a responsibility which must be borne by all sectors of society. Of course, this starts in ordinary homes where the family, the fundamental pillar of society, strives to transmit traditional and sound values to children by a deliberate effort to inform their consciences. The younger generations deserve and indeed thrive upon value-based education which reinforces respect, acceptance, compassion and equality. It is important therefore that schools, both government and faith-based, do all possible to support parents in the delicate but satisfying task of raising children to appreciate all that is good and true.6. The media's power to shape minds, especially of the young, cannot be underestimated. While the irresponsible elements within it are increasingly being recognized for what they are, it is also the case that much good can be effected through quality productions and educational programmes. When people working within the media exercise their moral conscience, it is possible to dispel ignorance and impart knowledge, preserve social values, and portray the transcendental dimension of life which arises from the spiritual nature of all people. Religious believers serve society admirably by collaborating in such projects for the common good.7. Ultimately, the aim of true education is to bring the individual to encounter the ultimate purpose of life. This motivates the person to serve broken humanity. Together may we continue to contribute toward peace and harmony in our society and the world. We Catholics join you with our heartfelt greetings as you celebrate this feast and I wish you once again a happy Vesakh. [End Page 132]Paul Cardinal Poupard, President and Archbishop Pier Luigi Celata, SecretaryPontifical Council for Interreligious DialogueCopyright © 2007 The University of Hawai'i Press... (shrink)

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

Karol Wojtyła found phenomenology very helpful for the analysis of concrete human experience and for overcoming the ethical formalism ofKant. Phenomenology, he believed, could also enrich classical Thomism by exploring the lived experience of freedom, interiority, and self-governance. But phenomenology, in his opinion, needed to be supplemented by metaphysics in order to ground experiences such as the sense of duty in the real order. He criticized much modern philosophy for abandoning metaphysics and thus neglecting the sapiential dimension. Since his career (...) as a professor was very short, he did not have time to complete his project of a personalist Thomism in which phenomenology and metaphysics would be harmoniously combined. (shrink)

One of the few points of unquestioned agreement in virtue theory is that the virtues are supposed to be excellences. The best way to understand the project of "virtue ethics" is to understand this claim as the idea that the virtues always yield correct moral action and, therefore, that we cannot be “too virtuous”. In other words, the virtues cannot be had in excess or “to a fault”. If we take this seriously, however, it yields the surprising conclusion that many (...) traits which have been traditionally thought of as “virtues” fail to make the grade. The most prominent solution to the problem, reminiscent of Aristotle's view, is found to generate more problems than it solves. -/- Ratio, Volume 35, Issue 1, Page 49-60, March 2022. (shrink)

Versions of Laver sequences are known to exist for supercompact and strong cardinals. Assuming very strong axioms of infinity, Laver sequences can be constructed for virtually any globally defined large cardinal not weaker than a strong cardinal; indeed, under strong hypotheses, Laver sequences can be constructed for virtually any regular class of embeddings. We show here that if there is a regular class of embeddings with critical point κ, and there is an inaccessible above κ, then it is (...) consistent for there to be a regular class that admits no Laver sequence. We also show that extendible cardinals are Laver-generating, i.e., that assuming only that κ is extendible, there is an extendible Laver sequence at κ. We use the method of proof to answer a question about Laver-closure of extendible cardinals at inaccessibles. Finally, we consider Laver sequences for super-almost-huge cardinals. Assuming slightly more than super-almost-hugeness, we show that there are super-almost-huge Laver sequences, improving the previously known upper bound for such Laver sequences. We also describe conditions under which the canonical construction of a Laver sequence fails for super-almost-huge cardinals. (shrink)

What Numbers Could Not Be’) that an adequate account of the numbers and our arithmetic practice must satisfy not only the conditions usually recognized to be necessary: (a) identify some w-sequence as the numbers, and (b) correctly characterize the cardinality relation that relates a set to a member of that sequence as its cardinal number—it must also satisfy a third condition: the ‘<’ of the sequence must be recursive. This paper argues that adding this further condition was a mistake—any (...) w-sequence would do, no matter how undecidable its ‘<’ relation. (shrink)

The meaning of 'most' can be described in many ways. We offer a framework for distinguishing semantic descriptions, interpreted as psychological hypotheses that go beyond claims about sentential truth conditions, and an experiment that tells against an attractive idea: 'most' is understood in terms of one-to-one correspondence. Adults evaluated 'Most of the dots are yellow', as true or false, on many trials in which yellow dots and blue dots were displayed for 200 ms. Displays manipulated the ease of using a (...) 'one-to-one with remainder' strategy, and a strategy of using the Approximate Number System to compare of (approximations of) cardinalities. Interpreting such data requires care in thinking about how meaning is related to verification. But the results suggest that 'most' is understood in terms of cardinality comparison, even when counting is impossible. (shrink)

In 1970, K. Kunen, working in the context of Kelley–Morse set theory, showed that the existence of a nontrivial elementary embedding j:V→V is inconsistent. In this paper, we give a finer analysis of the implications of his result for embeddings V→V relative to models of ZFC. We do this by working in the extended language , using as axioms all the usual axioms of ZFC , along with an axiom schema that asserts that j is a nontrivial elementary embedding. Without (...) additional axiomatic assumptions on j, we show that that the resulting theory is weaker than an ω-Erdös cardinal, but stronger than n-ineffables. We show that natural models of ZFC+BTEE give rise to Schindler’s remarkable cardinals. The approach to inconsistency from ZFC+BTEE forks into two paths: extensions of ZFC+BTEE+Cofinal Axiom and ZFC+BTEE+¬Cofinal Axiom, where Cofinal Axiom asserts that the critical sequence is cofinal in the ordinals. We describe near-minimal inconsistent extensions of each of these theories. The path toward inconsistency from ZFC+BTEE+¬Cofinal Axiom is paved with a sequence of theories of increasing large cardinal strength. Indeed, the extensions of the theory ZFC +“j is a nontrivial elementary embedding” form a hierarchy of axioms, ranging in strength from Con to the existence of a cardinal that is super-n-huge for every n, to inconsistency. This hierarchy is parallel to the usual hierarchy of large cardinal axioms, and can be used in the same way. We also isolate several intermediate-strength axioms which, when added to ZFC+BTEE, produce theories having strengths in the vicinity of a measurable cardinal of high Mitchell order, a strong cardinal, ω Woodin cardinals, and n-huge cardinals. We also determine precisely which combinations of axioms, of the form result in inconsistency. (shrink)

Notre époque est émotionnelle : elle aime les sensations fortes, les défis délirants, la violence. Ces excès en tous genres, elle se les représente volontiers sous une forme extrême, où l'accent est mis sur ce qui chavire nos sens : l'intensité, la démesure, le moralement inadmissible, l'horreur. Un élan destructeur au point de rendre légitime, en termes spectaculaires, une " esthétique de la limite dépassée ". L'objet de ce livre est double. D'une part, documenter par le menu les formes d'expression (...) contemporaines fortes de ce désir de dépassement esthétique : spectacles superlatifs, performances artistiques engageant la souffrance, documents d'actualité insoutenables, images de la pornographie dure, cinéma violent, mises en scène de la scatophilie, idolâtrie de la mort et du cadavre. D'autre part, analyser le glissement vers l'esthétique extrême que consacre, plus qu'aucune autre, la société occidentale. Magnétique et médiatique, la représentation de l'extrême y constitue désormais une véritable culture, un nouveau référent, un but. Fourmillant d'exemples, cet essai s'attache enfin à apporter une réponse à ce questionnement cardinal : quel avenir, en Occident, pour le spectacle du pire? Car à l'esthétique extrême il y a, en bout de course, une conséquence dramatique : l'épuisement du désir de voir. Que faire dès lors pour revivifier ce désir sinon, à plus ou moins court terme, devoir extrêmiser l'extrême lui-même et ses représentations? (shrink)

Versions of Laver sequences are known to exist for supercompact and strong cardinals. Assuming very strong axioms of infinity, Laver sequences can be constructed for virtually any globally defined large cardinal not weaker than a strong cardinal; indeed, under strong hypotheses, Laver sequences can be constructed for virtually any regular class of embeddings. We show here that if there is a regular class of embeddings with critical point $\kappa$, and there is an inaccessible above $\kappa$, then it is (...) consistent for there to be a regular class that admits no Laver sequence. We also show that extendible cardinals are Laver-generating, i.e., that assuming only that $\kappa$ is extendible, there is an extendible Laver sequence at $\kappa$. We use the method of proof to answer a question about Laver-closure of extendible cardinals at inaccessibles. Finally, we consider Laver sequences for super-almost-huge cardinals. Assuming slightly more than super-almost-hugeness, we show that there are super-almost-huge Laver sequences, improving the previously known upper bound for such Laver sequences. We also describe conditions under which the canonical construction of a Laver sequence fails for super-almost-huge cardinals. (shrink)

In this paper we introduce the Wholeness Axiom , which asserts that there is a nontrivial elementary embedding from V to itself. We formalize the axiom in the language {∈, j } , adding to the usual axioms of ZFC all instances of Separation, but no instance of Replacement, for j -formulas, as well as axioms that ensure that j is a nontrivial elementary embedding from the universe to itself. We show that WA has consistency strength strictly between I 3 (...) and the existence of a cardinal that is super- n -huge for every n . ZFC + WA is used as a background theory for studying generalizations of Laver sequences. We define the notion of Laver sequence for general classes E consisting of elementary embeddings of the form i :V β → M , where M is transitive, and use five globally defined large cardinal notions – strong, supercompact, extendible, super-almost-huge, superhuge – for examples and special cases of the main results. Assuming WA at the beginning, and eventually refining the hypothesis as far as possible, we prove the existence of a strong form of Laver sequence for a broad range of classes E that include the five large cardinal types mentioned. We show that if κ is globally superstrong, if E is Laver-closed at beth fixed points, and if there are superstrong embeddings i with critical point κ and arbitrarily large targets such that E is weakly compatible with i , then our standard constructions are E -Laver at κ . , there is an extendible Laver sequence at κ .) In addition, in most cases our Laver sequences can be made special if E is upward λ -closed for sufficiently many λ. (shrink)

We present the results of four experiments concerning the evaluation people make of sentences involving “many”, showing that two sentences of the form “many As are Bs” vs. “many As are Cs” need not be equivalent when evaluated relative to a background in which B and C have the same cardinality and proportion to A, but in which B and C are predicates with opposite semantic and affective values. The data provide evidence that subjects lower the standard relevant to ascribe (...) “many” for the more negative predicate, and that judgments involving “many” are sensitive to moral considerations, namely to expectations involving a representation of the desirability as opposed to the mere probability of an outcome. We relate the results to similar semantic asymmetries discussed in the psychological literature, in particular to the Knobe effect and to framing effects. (shrink)

Three theories contend as explanations of perpetrator behavior in the Holocaust as well as other cases of genocide: structural, intentional, and situational. Structural explanations emphasize the sense in which no single individual or choice accounts for the course of events. In opposition, intentional/cutltural accounts insist upon the genocides as intended outcomes, for how can one explain situations in which people ‘step up’ and repeatedly kill defenseless others in large numbers over sustained periods of time as anything other than a choice? (...) Situational explanations offer a type of behavioral account; this is how people act in certain environments. Critical to the situational account as I discuss it is the ‘Asch paradigm’, i.e. experimentally attested conditions for eliciting conformityof behavior regardlesss of available evidence of prior beliefs. In what follows, I defend what I term above a version of situational explanations of perpetrator behavior. Moreover, I maintain that the factors that explain provide an understanding as well. While not committed to the complete irrelevance or exclusion of cultural or structural factors, nonetheless situational analyses can account both for what happened and why. A cardinal virtue of this version of situational explanations consists in showing how shallow the problem of understanding turns out to be for such cases. (shrink)

The monthly magazine Hochland was probably the most influential Catholic cultural periodical in Germany in the Weimar Period. According to Georg Cardinal von Kopp’s assessment in 1911, it was “unfortunately the most read periodical in all of the educated circles of Germany, Austria and German Switzerland”. Moving beyond the simple rejection of modern culture in Germany, the journal tried to follow a new program of mediatory engagement, although it did continue to hold to traditional positions in many regards. In (...) this article the reception of modern, Enlightenment-affirmative philosophy of religion in the journal is introduced with reference to reviews and essays from the later 1910s to the early 1930s. The journal’s treatment of a few critical subject areas is given close interpretive analysis, including the journal’s treatment of Gertrud Simmel’s Über das Religiöse, individually conceptualized forms of personalist moral theory, and the general shift to phenomenological discourses and the individual in the philosophy of religion. The fundamental rejections of these ideas and these schools of thought in reviews and essays, which are also found in the journal at this time (as in most all German language Catholic cultural journals of the period), are not addressed in this article. The article thus sheds light on an often-forgotten and relatively small minority phenomenon in German Catholic intellectual circles of the Weimar Period, namely the positive embrace of Enlightenment-oriented modern thought. By promoting these ideas at this time, this group made themselves highly vulnerable to disciplinary measures by the Catholic Church. (The journal was put on the Index in 1911.). (shrink)

The monthly magazine Hochland was probably the most influential Catholic cultural periodical in Germany in the Weimar Period. According to Georg Cardinal von Kopp’s assessment in 1911, it was “unfortunately the most read periodical in all of the educated circles of Germany, Austria and German Switzerland”. Moving beyond the simple rejection of modern culture in Germany, the journal tried to follow a new program of mediatory engagement, although it did continue to hold to traditional positions in many regards. In (...) this article the reception of modern, Enlightenment-affirmative philosophy of religion in the journal is introduced with reference to reviews and essays from the later 1910s to the early 1930s. The journal’s treatment of a few critical subject areas is given close interpretive analysis, including the journal’s treatment of Gertrud Simmel’s Über das Religiöse, individually conceptualized forms of personalist moral theory, and the general shift to phenomenological discourses and the individual in the philosophy of religion. The fundamental rejections of these ideas and these schools of thought in reviews and essays, which are also found in the journal at this time (as in most all German language Catholic cultural journals of the period), are not addressed in this article. The article thus sheds light on an often-forgotten and relatively small minority phenomenon in German Catholic intellectual circles of the Weimar Period, namely the positive embrace of Enlightenment-oriented modern thought. By promoting these ideas at this time, this group made themselves highly vulnerable to disciplinary measures by the Catholic Church. (The journal was put on the Index in 1911.). (shrink)

In this original and compelling book, Jeffrey P. Bishop, a philosopher, ethicist, and physician, argues that something has gone sadly amiss in the care of the dying by contemporary medicine and in our social and political views of death, as shaped by our scientific successes and ongoing debates about euthanasia and the "right to die"--or to live. __The Anticipatory Corpse: Medicine, Power, and the Care of the Dying__, informed by Foucault's genealogy of medicine and power as well as by a (...) thorough grasp of current medical practices and medical ethics, argues that a view of people as machines in motion--people as, in effect, temporarily animated corpses with interchangeable parts--has become epistemologically normative for medicine. The dead body is subtly anticipated in our practices of exercising control over the suffering person, whether through technological mastery in the intensive care unit or through the impersonal, quasi-scientific assessments of psychological and spiritual "medicine." The result is a kind of nihilistic attitude toward the dying, and troubling contradictions and absurdities in our practices. Wide-ranging in its examples, from organ donation rules in the United States, to ICU medicine, to "spiritual surveys," to presidential bioethics commissions attempting to define death, and to high-profile cases such as Terri Schiavo's, __The Anticipatory Corpse__ explores the historical, political, and philosophical underpinnings of our care of the dying and, finally, the possibilities of change. A ground-breaking work in bioethics, this book will provoke thought and argument for all those engaged in medicine, philosophy, theology, and health policy. "With extraordinary philosophical sophistication as well as knowledge of modern medicine, Bishop argues that the body that shapes the work of modern medicine is a dead body. He defends this claim decisively with with urgency. I know of no book that is at once more challenging and informative as __The Anticipatory Corpse. __To say this book is the most important one written in the philosophy of medicine in the last twenty-five years would not do it justice. This book is destined to change the way we think and, hopefully, practice medicine." --_Stanley Hauerwas, Duke Divinity School _ "Jeffrey Bishop carefully builds a detailed, scholarly case that medicine is shaped by its attitudes toward death. Clinicians, ethicists, medical educators, policy makers, and administrators need to understand the fraught relationship between clinical practices and death, and __The Anticipatory Corpse __is an essential text. Bishop's use of the writings of Michel Foucault is especially provocative and significant. This book is the closest we have to a genealogy of death." --_Arthur W. Frank, University of Calgary _ "Jeffrey Bishop has produced a masterful study of how the living body has been placed within medicine's metaphysics of efficient causality and within its commitment to a totalizing control of life and death, which control has only been strengthened by medicine's taking on the mantle of a bio-psycho-socio-spiritual model. This volume's treatment of medicine's care of the dying will surely be recognized as a cardinal text in the philosophy of medicine." --_H. Tristram Engelhardt, Jr., Rice University, Baylor College of Medicine_. (shrink)

In set theory without the Axiom of Choice, we investigate the set-theoretic strength of the principle NDS which states that there is no function f on the set ω of natural numbers such that for everyn ∈ ω, f ≺ f, where for sets x and y, x ≺ y means that there is a one-to-one map g : x → y, but no one-to-one map h : y → x. It is a long standing open problem whether NDS implies (...) AC. In this paper, among other results, we show that NDS is a strong axiom by establishing that ACLO ↛ NDS in ZFA set theory. The latter result provides a strongly negative answer to the question of whether “every Dedekind-finite set is finite” implies NDS addressed in G. H. Moore “Zermelo’s Axiom of Choice. Its Origins, Development, and Influence” and in P. Howard–J. E. Rubin “Consequences of the Axiom of Choice”. We also prove that ACWO ↛ NDS in ZF and that “for all infinite cardinals m, m + m = m” ↛ NDS in ZFA. (shrink)

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

Assuming the existence of a supercompact limit of supercompact cardinals, we modify the original consistency proof of Martin's Maximum to obtain a model in which MM holds but the axiom fails.

We suggest a new approach for addressing the problem of establishing an axiomatic foundation for large cardinals. An axiom asserting the existence of a large cardinal can naturally be viewed as a strong Axiom of Infinity. However, it has not been clear on the basis of our knowledge of ω itself, or of generally agreed upon intuitions about the true nature of the mathematical universe, what the right strengthening of the Axiom of Infinity is—which large cardinals ought to be (...) derivable? It was shown in the 1960s by Lawvere that the existence of an infinite set is equivalent to the existence of a certain kind of structure-preserving transformation from V to itself, not isomorphic to the identity. We use Lawvere's transformation, rather than ω, as a starting point for a reasonably natural sequence of strengthenings and refinements, leading to a proposed strong Axiom of Infinity. A first refinement was discussed in later work by Trnková—Blass, showing that if the preservation properties of Lawvere's tranformation are strengthened to the point of requiring it to be an exact functor , such a transformation is provably equivalent to the existence of a measurable cardinal. We propose to push the preservation properties as far as possible, short of inconsistency. The resulting transformation V→V is strong enough to account for virtually all large cardinals, but is at the same time a natural generalization of an assertion about transformations V→V known to be equivalent to the Axiom of Infinity. (shrink)

L'oeuvre du philosophe Paul Gilbert, jésuite belge professeur de métaphysique depuis vingt-cinq ans à l'Université Grégorienne, est considérable et mérite d'être mieux connue. Après avoir étudié les écrits de saint Anselme (Dire l'ineffable, 1984), Paul Gilbert a questionné la métaphysique classique, sous l'influence de la philosophie réflexive (La Simplicité du principe, 1994). Il a ensuite souligné les chances offertes à la réflexion fondamentale par l'analogie, thème classique mais oublié (La Patience d'être, 1996, et Sapere e sperare, 2003). Plus (...) récemment, il a abordé la question de la violence, ce mal- être si commun (Violence et compassion, 2009). En 1997 déjà, son travail a été récompensé du prix Cardinal Mercier de l'U.C.Louvain-la-Neuve. Durant de nombreuses années, Paul Gilbert a été professeur invité à l'Institut catholique de Paris. A l'occasion de son éméritat dans cette institution, un colloque y a été organisé. Dans leurs communications, S. Bonanni, Ph. Capelle-Dumont, S. D'Agostino, E. Gabellieri, J. de Gramont, 1. Malaguti, R. Repole et E. Tourpe reprennent et développent l'élaboration de la pensée de Paul Gilbert. Ce dernier fait écho à chacune d'elles et, dans un exposé final, retrace son propre itinéraire intellectuel. (shrink)

We show that the members of a certain class of semi-proper iterations do not add countable sets of ordinals. As a result, starting from suitable large cardinals one can obtain a model in which the Continuum Hypothesis holds and every function from ω1 to ω1 is bounded on a club by a canonical function for an ordinal less than ω2.

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

Working in the context of restricted forms of the Axiom of Choice, we consider the problem of splitting the ordinals below λ of cofinality θ into λ many stationary sets, where θ < λ are regular cardinals. This is a continuation of [4].

This line has long been a crux in the interpretation of Pindar, and there is still no consensus on its syntax or meaning. The conclusions reached by Stefan Radt , 148–74) and Richard Stoneman , 65–70) in the most recent studies of the problem are in all respects at variance. The cardinal difficulty of0 the line is the sense of , which must be elucidated before one can attempt to disentangle the syntax. I believe that previous commentators have overlooked (...) or misapplied crucial parallel passages because of their preconceptions about the nature of the metaphor here. (shrink)

Two Fraenkel-Mostowski models are constructed in which the Boolean Prime Ideal Theorem is true. In both models, AC for countable sets is true, but AC for sets of cardinality 2math image and the 2m = m principle are both false. The Principle of Dependent Choices is true in the first model, but false in the second.

In Zermelo-Fraenkel set theory with the axiom of choice every set has the same cardinal number as some ordinal. Von Rimscha has weakened this condition to “Every set has the same cardinal number as some transitive set”. In set theory without the axiom of choice, we study the deductive strength of this and similar statements introduced by von Rimscha.

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

We study theorems from Functional Analysis with regard to their relationship with various weak choice principles and prove several results about them: “Every infinite‐dimensional Banach space has a well‐orderable Hamel basis” is equivalent to ; “ can be well‐ordered” implies “no infinite‐dimensional Banach space has a Hamel basis of cardinality ”, thus the latter statement is true in every Fraenkel‐Mostowski model of ; “No infinite‐dimensional Banach space has a Hamel basis of cardinality ” is not provable in ; “No infinite‐dimensional (...) Banach space has a well‐orderable Hamel basis of cardinality ” is provable in ; (the Axiom of Choice for denumerable families of non‐empty finite sets) is equivalent to “no infinite‐dimensional Banach space has a Hamel basis which can be written as a denumerable union of finite sets”; Mazur's Lemma (“If X is an infinite‐dimensional Banach space, Y is a finite‐dimensional vector subspace of X, and, then there is a unit vector such that for all and all scalars α”) is provable in ; “A real normed vector space X is finite‐dimensional if and only if its closed unit ball is compact” is provable in ; (Principle of Dependent Choices) + “ can be well‐ordered” does not imply the Hahn‐Banach Theorem () in ; and “no infinite‐dimensional Banach space has a Hamel basis of cardinality ” are independent from each other in ; “No infinite‐dimensional Banach space can be written as a denumerable union of finite‐dimensional subspaces” lies in strength between (the Axiom of Countable Choice) and ; implies “No infinite‐dimensional Banach space can be written as a denumerable union of closed proper subspaces” which in turn implies ; “Every infinite‐dimensional Banach space has a denumerable linearly independent subset” is a theorem of, but not a theorem of ; and “Every infinite‐dimensional Banach space has a linearly independent subset of cardinality ” implies “every Dedekind‐finite set is finite”. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Christ's Male Sexuality and Acting In Persona Christi:A New Argument in Favor of the All-Male PriesthoodPaul Gondreau"One must be allowed to think about and discuss the issues.... [And on the issue of women's ordination] the discussion is still with us, it is still alive, and cannot be stifled [ersticken] by a paper [ein Papier]." So declares Archbishop Stefan Hesse of Hamburg, Germany, in the summer of 2020, where "a (...) paper" refers to an apostolic letter, Pope St. John Paul II's 1994 Ordinatio Sacerdotalis, which definitively rules out the admissibility of women to priestly ordination.1 No doubt the archbishop was drawing inspiration from Cardinal Archbishop Reinhard Marx of Munich and Freising, who a month earlier had opined that arguments against women's ordination were becoming increasingly "weaker" (schwächer).2 In his capacity as president of the German bishops' conference, Bishop Georg Bätzing [End Page 805] of Limburg also pronounced on the issue in even more unabated terms; after insisting in June of the same year that Catholics can "continue to talk about the issue of women's ordination," he went further in an interview with the German magazine Herder Korrespondenz six months later: "There are well-developed arguments in theology in favor of opening up the sacramental ministry to women," he asserted, after declaring his desire for "change" (Veränderung), a change that he opined could begin with ordaining women as deacons before ordaining them as priests and bishops.3 Two years later, Bätzing remained adamant, going so far as to declare to speak in the name of the faith of the entire Church on this issue: "The sensus fidelium, that is, the sense of the faithful," he stated, "has moved on" (geht weiter).4 Since the deposit of faith, of which the sensus fidelium is expressive, does not and cannot "move on," that is, change, Bätzing's appeal to the sensus fidelium marks a curious—if also audacious—move, to say the least.It appears we are destined ever "to think about and discuss" the issue of women's ordination, no matter if the Holy Father himself, in the person of the John Paul II, should seek to quell this discussion in the strongest and most authoritative of terms: "In order that all doubt be removed regarding [this] matter," writes the Polish Pontiff, "I declare that the Church has no authority whatsoever to confer priestly ordination on women and that this judgment is to be definitively held by all the Church's faithful" (Ordinatio Sacerdotalis §4).As regards John Paul II's assertion that his judgment is to be "definitively held" by all Catholics, it is worth recalling that, even if one high-ranking German churchman (Hesse) might esteem Ordinatio Sacerdotalis as [End Page 806] tantamount to ein Papier das erstickt, "a paper that stifles," another, Joseph Ratzinger (in his role as cardinal prefect of the Congregation for the Doctrine of the Faith [CDF], before becoming Pope Benedict XVI), holds up John Paul II's apostolic letter as expressive of the infallible teaching of the Church's ordinary magisterium and as "belonging to the deposit of faith." As such, the letter requires all Catholics, asserts Ratzinger, to offer their "definitive assent" to the Holy Father's pronouncement.5John Paul II and Ratzinger may as well have been talking to the wind. If Archbishop Hesse and his disparaging attitude—dismissing a papal apostolic letter as a paper that "stifles"—serves to indicate what "definitive assent" looks like, and if Bishop Bätzing can find that the deposit of faith (represented by the sensus fidelium), particularly as it concerns women's ordination, is able to "move on," it is no wonder that we find the issue "still alive." And recall Hesse and Bätzing hardly stand alone among German prelates in their willingness to ignore Ordinatio Sacerdotalis, as Cardinal Marx and the much touted German "synodal way" show (the archbishop of Luxembourg, Cardinal Jean-Claude Hollerich, could also be added to the list).6 Further, when Hesse asserts, "one must be allowed to think about and discuss [women's ordination]," there is little doubt that, for him and those... (shrink)