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

The purpose of this valuable book is to consider recent cultural trends in bioethics from a Catholic perspective. Bioethics is intended for a lay audience interested in understanding bioethical issues from a Catholic perspective.

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

An important application of ultrafilters is in the ultraproduct construction in model theory. In this paper we study ultraproducts of countable structures, whose universe we assume is ω , using ultrafilters on a countable index set, which we also assume to be ω . Many of the properties of the ultraproduct are in fact inherent properties of the ultrafilter. For example, if we take a sequence of countable linear orders without maximal element, then their ultraproduct will have no maximal element, (...) and we can ask what its cofinality is. This cardinal depends only on the ultrafilter; it does not depend on what linear orders comprise the factors. (shrink)

We investigate the pair-splitting number $\germ{s}_{pair}$ which is a variation of splitting number, pair-reaping number $\germ{r}_{pair}$ which is a variation of reaping number and cardinal invariants of ideals on ω. We also study cardinal invariants of F σ ideals and their upper bounds and lower bounds. As an application, we answer a question of S. Solecki by showing that the ideal of finitely chromatic graphs is not locally Katětov-minimal among ideals not satisfying Fatou's lemma.

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)

There are long-standing doubts about whether data from subjective scales—for instance, self-reports of happiness—are cardinally comparable. It is unclear how to assess whether these doubts are justified without first addressing two unresolved theoretical questions: how do people interpret subjective scales? Which assumptions are required for cardinal comparability? This paper offers answers to both. It proposes an explanation for scale interpretation derived from philosophy of language and game theory. In short: conversation is a cooperative endeavour governed by various maxims (Grice 1989); (...) because subjective scales are vague and individuals want to make themselves understood, scale interpretation is a search for a focal point (Schelling 1960). A specific focal point it hypothesised; if this hypothesis is correct, subjective data will be cardinally comparable. Four individually necessary and jointly sufficient conditions for cardinal comparability are specified. The paper then argues this hypothesis can be empirically be tested, makes an initial attempt to do so using subjective well-being data, and concludes it is supported. Numerous areas for further research are identified including, at the end of the paper, how certain tests could be used to ‘correct’ subjective data if they are not cardinal. (shrink)

The three cardinal aims of science are prediction, control, and explanation; but the greatest of these is explanation. Also the most inscrutable: prediction aims at truth, and control at happiness, and insofar as we have some independent grasp of these notions, we can evaluate science’s strategies of prediction and control from the outside. Explanation, by contrast, aims at scientific understanding, a good intrinsic to science and therefore something that it seems we can only look to science itself to explicate.

I defend the idea that a liberal commitment to value neutrality is best honoured by maintaining a pure cardinality component in our rankings of opportunity or liberty sets. I consider two challenges to this idea. The first holds that cardinality rankings are unnecessary for neutrality, because what is valuable about a set of liberties from a liberal point of view is not its size but rather its variety. The second holds that pure cardinality metrics are insufficient for neutrality, because liberties (...) cannot be individuated into countable entities without presupposing some relevantly partisan evaluative perspective. I argue that a clear understanding of the liberal basis for valuing liberty shows the way to satisfying responses to both challenges. (shrink)

Abstract.This paper is the first in a series of three which culminates in an ordinal analysis of Π12-comprehension. On the set-theoretic side Π12-comprehension corresponds to Kripke-Platek set theory, KP, plus Σ1-separation. The strength of the latter theory is encapsulated in the fact that it proves the existence of ordinals π such that, for all β>π, π is β-stable, i.e. Lπ is a Σ1-elementary substructure of Lβ. The objective of this paper is to give an ordinal analysis of a scenario of (...) not too complicated stability relations as experience has shown that the understanding of the ordinal analysis of Π12-comprehension is greatly facilitated by explicating certain simpler cases first.This paper introduces an ordinal representation system based on ν-indescribable cardinals which is then employed for determining an upper bound for the proof–theoretic strength of the theory KPi+ ∀ρ ∃ππ is π+ρ-stable, where KPi is KP augmented by the axiom saying that every set is contained in an admissible set. (shrink)

In ordinal analysis of impredicative theories so-called collapsing functions are of central importance. Unfortunately, the definition procedure of these functions makes essential use of uncountable cardinals whereas the notation system that they call into being corresponds to a recursive ordinal. It has long been claimed that, instead, one should manage to develop such functions directly on the basis of admissible ordinals. This paper is meant to show how this can be done. Interpreting the collapsing functions as operating directly on admissible (...) sets also renders a new and perspicuous approach to well-ordering proofs possible. MSC: 03F15, 03F35. (shrink)

Let C be a small Barr-exact category, Reg the category of all regular functors from C to the category of small sets. A form of M. Barr's full embedding theorem states that the evaluation functor e : C →[Reg, Set ] is full and faithful. We prove that the essential image of e consists of the functors that preserve all small products and filtered colimits. The concept of κ-Barr-exact category is introduced, for κ any infinite regular cardinal, and the natural (...) generalization to κ-Barr-exact categories of the above result is proved. The treatment combines methods of model theory and category theory. Some applications to module categories are given. (shrink)

We define a rich model to be one which contains a proper elementary substructure isomorphic to itself. Existence, nonstructure, and categoricity theorems for rich models are proved. A theory T which has fewer than $\min(2^\lambda,\beth_2)$ rich models of cardinality $\lambda(\lambda > |T|)$ is totally transcendental. We show that a countable theory with a unique rich model in some uncountable cardinal is categorical in ℵ 1 and also has a unique countable rich model. We also consider a stronger notion of richness, (...) and use it to characterize superstable theories. (shrink)

An ordering (≤K) on maximal almost disjoint (MAD) families closely related to destructibility of MAD families by forcing is introduced and studied. It is shown that the order has antichains of size.

Moral behaviour, and more recently wisdom and prudence, are emerging as areas of interest in the study of business ethics and management. The purpose of this article is to illustrate that Cicero—lawyer, politician, orator and prolific writer, and one of the earliest experts in the field recognised the significance of moral behaviour in his society. Cicero wrote ‘Moral Duties’ (De Officiis) about 44 BC. He addressed the four cardinal virtues wisdom, justice, courage and temperance, illustrating how practical wisdom, theoretical/conceptual wisdom (...) and justice were viewed in Rome of the first century BC. ‘Moral Duties’ is a letter admonishing his son, Marcus. It refers to personal behaviour, business practice and analyses terms such as good faith and criminal fraud. In addition, it contains material which would be suitable for tutorials/seminars and discussions, particularly in the areas of critical thinking in business ethics and general management. A study of De Officiis in respect to present day management and business practice could give a wider perspective to business ethics and management students. If concepts such as moral virtue, moral propriety and moral goodness, many of which seem to be ignored in business situations today, are to be embedded in business leaders of the future, it is reasonable to expect that these qualities will be analysed and discussed by business students today. Further, a study of Cicero’s six-step approach, when preparing an address/speech, could be useful and productive for practitioners and students in this area. (shrink)

We study the possible values of the cofinality invariant for various Borel ideals on the natural numbers. We introduce the notions of a fragmented and gradually fragmented ideal and prove a dichotomy for fragmented ideals. We show that every gradually fragmented ideal has cofinality consistently strictly smaller than the cardinal invariant and produce a model where there are uncountably many pairwise distinct cofinalities of gradually fragmented ideals.

We introduce a family of rank functions and related notions of total transcendence for Galois types in abstract elementary classes. We focus, in particular, on abstract elementary classes satisfying the condition known as tameness, where the connections between stability and total transcendence are most evident. As a byproduct, we obtain a partial upward stability transfer result for tame abstract elementary classes stable in a cardinal $\lambda$ satisfying $\lambda^{\aleph_{0}}\gt \lambda$, a substantial generalization of a result of Baldwin, Kueker, and VanDieren.

Michael Novak's eyewitness report on the second and pivotal session of Vatican II in 1964 vividly inter weaves pageantry, politics, and theology. An unusually well-informed lay intellectual, who had earned a theological degree just before the Council, Novak applauded the purposes of Pope John XXIII and his successor Paul VI-"to throw open the windows of the church." In this report, he coined the classic description of the foes of the reforms at Vatican II as the party of "nonhistorical orthodoxy," (...) emphasizing the eternal and unchanging, neglecting history and contingency. The author recounts many moments of high drama-Pope Paul VI's opening speech, the vote on the collegiality of bishops, the plea of Cardinal Bea on behalf of the chapter on Jews, and Bishop De Smedt's defense of religious freedom. His colorful chapter on the American bishops in 1964 serves as a fascinating benchmark, as do his many insights into the new role of the laity. His final chapter is a moving tribute to the Open Church engaging the contemporary world, and his new introduction brings this report up to date. This work will be of compelling interest to those interested in the post-conciliar fall of Communism, under the great John Paul II-who took his name from his two predecessors at Vatican II. The winner of the million-dollar Templeton Prize for Progress in Religion, Michael Novak is a theologian, author, and former U.S. ambassador. He currently holds the George Frederick Jewett Chair in Religion and Public Policy at the American Enterprise Institute in Washington, D.C. where he is director of social and political studies. His writings have appeared in every major Western language, and in Chinese, Bengali, Korean, and Japanese. Also available from Transaction are his Catholic Social Thought and Liberal Institutions, The Experience of Nothingness, The Guns of Lattimer, Unmeltable Ethnics, Belief and Unbelief, and Choosing Presidents. (shrink)

This paper considers a range of infinite exchange problems, including one recent example discussed by Barrett and Arntzenius, and propose a general taxonomy based on cardinality considerations and the possibility of identifying and tracking the units of exchange.

Many Sides is the first full-length study of Protagorean antilogic, an argumentative practice with deep roots in rhetorical history and renewed relevance for contemporary culture. Founded on the philosophical relativism of Protagoras, antilogic is a dynamic rather than a formal approach to argument, focused principally on the dialogical interaction of opposing positions (anti-logoi) in controversy. In ancient Athens, antilogic was the cardinal feature of Sophistic rhetoric. In Rome, Cicero redefined Sophistic argument in a concrete set of dialogical procedures. In turn, (...) Quintilian inherited this dialogical tradition and made it the centrepiece of his own rhetorical practice and pedagogy. Many Sides explores the history, theory, and pedagogy of this neglected rhetorical tradition and, by appeal to recent rhetorical and philosophical theory, reconceives the enduring features of antilogical practice in a dialogical approach to argumentation especially suited to the pluralism of our own age and the diversity of modern classrooms. (shrink)

Following the publication of Pope Francis’ Apostolic Exhortation Amoris Laetitia, four aged Cardinals wrote to him asking him to clarify positions in the Exhortation they charged were causing confusion to the faithful. They even suggested he had changed some Catholic doctrines. This essay answers their questions, arguing that Francis has not changed any Catholic doctrine but has changed, in the sense that he has renewed, Catholic pastoral practice. It also argues that, while not changing any Catholic doctrines, he has reprioritized (...) some of them, especially the doctrines of the authority and inviolability of conscience and what he calls the “irregular situations” of cohabitation, same-sex unions, and divorce and remarriage without annulment. (shrink)

We examine the question of how many Boolean algebras, distinct up to isomorphism, that are quotients of the powerset of the naturals by Borel ideals, can be proved to exist in ZFC alone. The maximum possible value is easily seen to be the cardinality of the continuum $2^{\aleph_{0}}$ ; earlier work by Ilijas Farah had shown that this was the value in models of Martin's Maximum or some similar forcing axiom, but it was open whether there could be fewer in (...) models of the Continuum Hypothesis. We develop and apply a new technique for constructing many ideals whose quotients must be nonisomorphic in any model of ZFC. The technique depends on isolating a kind of ideal, called shallow, that can be distinguished from the ideal of all finite sets even after any isomorphic embedding, and then piecing together various copies of the ideal of all finite sets using distinct shallow ideals. In this way we are able to demonstrate that there are continuum-many distinct quotients by Borel ideals, indeed by analytic P-ideals, and in fact that there is in an appropriate sense a Borel embedding of the Vitali equivalence relation into the equivalence relation of isomorphism of quotients by analytic P-ideals. We also show that there is an uncountable definable wellordered collection of Borel ideals with distinct quotients. (shrink)

We investigate the following weak Ramsey property of a cardinal κ: If χ is coloring of nodes of the tree κ <ω by countably many colors, call a tree ${T \subseteq \kappa^{ < \omega}}$ χ-homogeneous if the number of colors on each level of T is finite. Write ${\kappa \rightsquigarrow (\lambda)^{ < \omega}_{\omega}}$ to denote that for any such coloring there is a χ-homogeneous λ-branching tree of height ω. We prove, e.g., that if ${\kappa < \mathfrak{p}}$ or ${\kappa > \mathfrak{d}}$ (...) is regular, then ${{\kappa \rightsquigarrow (\kappa)^{ < \omega}_{\omega}}}$ and that ${\mathfrak{b}}$ ${(\mathfrak{b})^{ < \omega}_{\omega}}$ and ${\mathfrak{d}}$ ${(\mathfrak{d})^{ < \omega}_{\omega}}$ . The arrow is applied to prove a generalization of a theorem of Hurewicz: A Čech-analytic space is σ-locally compact iff it does not contain a closed homeomorphic copy of irrationals. (shrink)

All of Mikhail Bakhtin’s work stands under the sign of plurality, the mystery of the one and the many. Unlike the third eye of Tibetan Buddhism, which gives those who possess it a vision of the secret unity holding creation together, Bakhtin seems to have had a third ear that permitted him to hear differences where others perceived only sameness, especially in the apparent wholeness of the human voice. The obsessive question at the heart of Bakhtin’s thought is always “Who (...) is talking?” It was his sense of the world’s overwhelming multiplicity that impelled Bakhtin to rethink strategies by which heterogeneity had traditionally been disguised as a unity. In his several attempts to find a single name for the teeming forces which jostled each other within the combat zone of the word—whether the term was “polyphony,” “heteroglossia,” or “speech communion”—Bakhtin was at great pains never to sacrifice the tension between identity and difference that fueled his enterprise. He always sought the minimum degree of homogenization necessary to any conceptual scheme, feeling it was better to preserve the heterogeneity which less patient thinkers found intolerable—and to which they therefore hurried to assign a unitizing label.Bakhtin’s metaphysical contrariness has the effect of making at times appear to be indiscriminate, as when he refused to recognize borders between biography and autobiography or, more notoriously, between speaking and writing. But, as I hope to show, these apparently cardinal distinctions are for Bakhtin only local instances of unity that participate in and are controlled by a fare more encompassing set of oppositions and differences. All this places an extra burden on those who seek an overarching design in Bakhtin’s legacy: the apparently unitizing term “Bakhtin” proves to be as illusory—or more illusory—in its ability to subsume real distinctions as any other, if we submit it to a Bakhtinian analysis. Michael Holquist is professor and chairman of the department of Slavic languages and literatures a Indiana University. With his wife, Katerina Clark, he has just completed Mikhail Bakhtin, a study of Bakhtin’s life and works, forthcoming in the autumn of 1984. He is currently working in Moscow. (shrink)

Mathematical objects, if they exist at all, exist independently of our proofs, constructions and stipulations. For example, whether inaccessible cardinals exist or not, the very act of our proving or postulating that they do doesn’t make it so. This independence thesis is a central claim of mathematical realism. It is also one that many anti-realists acknowledge too. For they agree that we cannot create mathematical truths or objects, though, to be sure, they deny that mathematical objects exist at all. I (...) have defended a mathematical realism of sorts. I interpret the objects of mathematics as positions in patterns, and maintain that they exist independently of us, and our stipulations, proofs, and the like. (shrink)

The Council of Trent and the Second Vatican Council are significant both to Lutheranism and Science. The first inaugurated the Counter Reformation and formulated a decree related to biblical hermeneutics later used as a basis for Galileo's condemnation. The second modernized the Roman Catholic Church and formulated the Pastoral Constitution Gaudium et spes used by Pope John Paul II as a basis for the reconsideration of the condemnation. In both cases, however, the Church of Rome may not have followed the (...) spirit of its councils. In the 17th century it condemned Galileo following a radical interpretation of the teaching of Trent, that of Cardinal Robert Bellarmine. Recently it reconsidered the ‚Affair’ unsatisfactorily probably because Gaudium et spes encouragement of science clashes with the possibility that Saint Bellarmine misinterpreted the Trent decree. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Buddhist-Christian Studies 23 (2003) 147-148 [Access article in PDF] Buddhists and Christians: Praying for Peace in the World Archbishop Michael L. Fitzgerald Pontifical Council for Interreligious Dialogue Dear Buddhist Friends:As the new president of the Pontifical Council for Interreligious Dialogue, the office of His Holiness the Pope for relations with people of different religious traditions, I wish to greet you and send this congratulatory message on the occasion (...) of the feast of Vesakh. This gesture of friendship, initiated in 1995 by my predecessor Cardinal Francis Arinze, has almost become a tradition. I wish to continue this good tradition and express my hearty congratulations to each and every one of you.In this message, I would like to invite you, my dear Buddhist friends, to join in prayer for the cause of peace in the world. Observing the current international situation, we cannot but be aware of the acuteness of the question of peace in our world. Since the beginning of this new millennium, marked by the dramatic events of September 11, 2001, we witness every day fresh scenes of bloodshed, violence, confrontation, and crisis in almost all parts of the world. In the midst of this grave situation, we cannot lead our lives without committing ourselves to advancing the cause of peace in the world.We Christians and Buddhists are convinced that the origin of all conflict is ultimately located in human hearts characterized by selfish desire, specifically by desire for power, domination, and wealth, often at the expense of others. It is also our common conviction that peace must inhabit people's hearts before it can become a social reality. For us, therefore, the most fundamental and efficient way to advance peace is to do our best to see that the deep-rooted selfishness of human hearts is overcome, so that people may be transformed into true artisans of peace.Pope John Paul II has proclaimed the year from October 2002 to October 2003 the Year of the Rosary of the Virgin Mary. He has earnestly encouraged the frequent recitation of the Rosary in order to pray for peace in the world. His wish to revive the practice of the Rosary is closely connected with the present historical circumstances, which need more than ever-constant supplication for the great gift of peace.My Buddhist friends, is it not a wonderful coincidence that you also have a lengthy tradition of using the Mala for prayer? The Rosary for Catholics and the [End Page 147] Mala for Buddhists are simple yet profound and meaningful prayer, despite essential differences in their form and content, based on our distinct doctrines and practices. For Catholics, the Rosary represents a most effective means of fostering contemplation of Jesus Christ. For Buddhists, the Mala is used to overcome the 108 Sinful Desires in order to reach the state of Nirvana. By virtue of their meditative character, these two prayers have in common a calming effect on those who pray them; they lead them to experience and to work for peace, and they produce fruits of love. For Catholics, the repetition and meditation of the holy names of the Persons of the Blessed Trinity and the Virgin Mary in the recitation of the Rosary makes us more willing to assimilate their love and compassion for others, especially for the poor and afflicted. In your Buddhist tradition, praying the Mala helps one to become a peacemaker.Dear Buddhist friends, these are the thoughts I wish to share with you this year. I am convinced that by persevering in prayer we will contribute to advancing peace in the world both now and in the future. May this peace be with you and your families on the feast of Vesakh and at all times.Archbishop Michael L. Fitzgerald, President © by Michael L. Fitzgerald. Reprinted by permission.... (shrink)

In this paper we show that it is consistent with ZFC that the cardinality of every maximal cofinitary group of Sym(ω) is strictly greater than the cardinal numbers o and a.

This paper begins by pointing out that the Aristotelian conception of continuity (synecheia) and the contemporary topological account share the same intuitive, proto-topological basis: the conception of a ?natural whole? or unity without joints or seams. An argument of Aristotle to the effect that what is continuous cannot be constituted of ?indivisibles? (e.g., points) is examined from a topological perspective. From that perspective, the argument fails because Aristotle does not recognize a collective as well as a distributive concept of a (...) multiplicity of points. It is the former concept that allows contemporary topology to identify some point sets with spatial regions (in the proto-topological sense of this term). This identification, in turn, allows contemporary topology to do what Aristotle was unwilling to do: to conceive the property of continuity, as well as the properties of having measure greater than zero and having n- dimension, as emergent properties. Thus, a point set can be continuous (connected) although none of its subsets of sufficiently smaller cardinality can be. Finally, the paper discusses the manner in which a topological principle, viz., the principle that none of the singletons of points of a continuum can be open sets of that continuum, captures certain aspects of the Aristotelian proto-topological conception of the relation between points and continua. E.g., for both Aristotle and contemporary topology, points in a continuum exist simple as limits of the remainder of the continuum: their singletons have empty ?interiors? and, hence, they are not ?chunks? (topologically, regular closed set) of the continuum. (shrink)

The overriding concern of theRepublicis the relation between morally right action and happiness. To settle the matter Plato says he must first discover the formula for politically just action for individuals, and that it will be easier to do this if we first take justice to be a virtue of a city as a whole. He develops a model of an ideal city, hardly mentioning justice and the other cardinal virtues, and then looks to that model to individuate them.Here I (...) shall first discuss the defense of his portrait of an ideal city. Second, the logic of virtueis considered, in two steps. It is argued that judgments of virtue are to be split into those of functionand power.Then there is a brief consideration of each half. Third, the implications of this for understanding and threefold classification of goods are drawn. Finally, there is a discussion of the formula for each of the cardinal virtues of a city. (shrink)

The overriding concern of theRepublicis the relation between morally right action and happiness. To settle the matter Plato says he must first discover the formula for politically just action for individuals, and that it will be easier to do this if we first take justice to be a virtue of a city as a whole. He develops a model of an ideal city, hardly mentioning justice and the other cardinal virtues, and then looks to that model to individuate them.Here I (...) shall first discuss the defense of his portrait of an ideal city. Second, the logic of virtueis considered, in two steps. It is argued that judgments of virtue are to be split into those of functionand power.Then there is a brief consideration of each half. Third, the implications of this for understanding and threefold classification of goods are drawn. Finally, there is a discussion of the formula for each of the cardinal virtues of a city. (shrink)

Probably the two most famous examples of elementary embeddings between inner models of set theory are the embeddings of the universe into an inner model given by a measurable cardinal and the embeddings of the constructible universeLinto itself given by 0#. In both of these examples, the “target model” is a subclass of the “ground model”. It is not hard to find examples of embeddings in which the target model is not a subclass of the ground model: ifis a generic (...) ultrafilter arising from forcing with a precipitous ideal on a successor cardinalκ, then the ultraproduct of the ground model viacollapsesκ. Such considerations suggest a classification of how close the target model comes to “fitting inside” the ground model.Definition 1.1. LetMandNbe inner models of ZFC, and letj:M→Nbe an elementary embedding. Theco-critical pointofjis the least ordinalλ, if any exist, such that there isX⊆λ, X∈NbutX∉M. Such anXis called anew subsetofλ.It is easy to see that the co-critical point ofj:M→Nis a cardinal inN. (shrink)

It is shown how the strong ordinal notation systems that figure in proof theory and have been previously defined by employing large cardinals, can be developed directly on the basis of their recursively large counterparts. Thereby we provide a completely new approach to well-ordering proofs as will be exemplified by determining the proof-theoretic ordinal of the systemKPM of [R91].

We broaden the framework of metric abstract elementary classes (mAECs) in several essential ways, chiefly by allowing the metric to take values in a well-behaved quantale. As a proof of concept we show that the result of Boney and Zambrano (Around the set-theoretical consistency of d-tameness of metric abstract elementary classes, arXiv:1508.05529, 2015) on (metric) tameness under a large cardinal assumption holds in this more general context. We briefly consider a further generalization to partial metric spaces, and hint at connections (...) to classes of fuzzy structures, and structures on sheaves. (shrink)