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

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)

This paper has two goals: 1) to understand justice as a cardinal virtue, according to Aquinas; and 2) to use his conception of justice as a cardinal virtue to understand how one engages in acts of “general” justice. The argument proceeds in four stages: 1) how Aquinas understands the virtues by looking to their “objects”; 2) the two distinct “modes” of the four cardinal virtues, as “general” and “specific” virtues; 3) the triangle of three kinds of justice, (...) seen in terms of their “objects”; 4) Aquinas’s doctrine of justice as a “general” virtue shows that we can perform operations of “general” justice in two ways, as do the ruler and his minsters, and as ordinary folk do. Surprisingly, it is the latter mode of acting for “general” justice that is primary, not the former. (shrink)

The notion of “ramifiability” , usually applied to cardinals, can be extended to directed sets and is put in relation here with familiar “large cardinal” properties.

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

Regularity is the thesis that all contingent propositions should be assigned probabilities strictly between zero and one. I will prove on cardinality grounds that if the domain is large enough, a regular probability assignment is impossible, even if we expand the range of values that probabilities can take, including, for instance, hyperreal values, and significantly weaken the axioms of probability.

The notions of finite and infinite second-order characterizability of cardinal and ordinal numbers are developed. Several known results for the case of finite characterizability are extended to infinite characterizability, and investigations of the second-order theory of ordinals lead to some observations about the Fraenkel-Carnap question for well-orders and about the relationship between ordinal characterizability and ordinal arithmetic. The broader significance of cardinal characterizability and the relationships between different notions of characterizability are also discussed.

Despite the recent claims of some prominent Catholic philosophers, I argue that Cardinal Newman's writings are in fact largely compatible with the contemporary movement in the philosophy of religion known as Reformed Epistemology, and in particular with the work of Alvin Plantinga. I first show how the thought of both Newman and Plantinga was molded in response to the "evidentialist" claims of John Locke. I then examine the details of Newman's response, especially as seen in his Essay in Aid (...) of A Grammar of Assent, suggesting that many of Newman's central ideas closely mirror Plantinga's. Finally, I argue that ifNewman and Plantinga part ways at any point, it is with respect to the basicality of specifically Christian (as opposed to theistic) belief. (shrink)

At the start of Early Modernity, the notion of raison d’état (reason of state) became a central issue in European politics. By means of that notion politicians and intellectuals reflected upon the legitimacy of violent and unethical means in the service of higher ends. Due, among others, to the nature of modern warfare, the role of the state acquired new significance in human affairs. Therefore, many philosophers and politicians argued that politics might be fundamentally different from other human activities and (...) might require a special rationality and morality. Also, thus my hypothesis, the discourse and history of the raison d’état is throughout history characterized by some recurring themes, such as the consolidation of governmental power. In this study, I want to explore how the raison d’état, in various strengths, has been instrumentalised in the political doctrine of Richelieu and Louis XIII as well in the political ideas of Putin and his ideological right-hand Dugin. It will be discussed how in both political systems; sovereignty is considered to be an undifferentiated body of legitimate power with a firm basis in Christian-Stoic natural law. The stoic ideal of indifference (apatheia) is an important reference in the reason of state rhetoric since the individual is supposed to be prepared to sacrifice his life for the preservation of the State. The common good should be the objective of each person in the society. I will analyse to what extent Putin and Dugin, in defending their war politics, use similar arguments as cardinal Richelieu. By reflecting on the intellectual discourse about the nature and purposes of the State itself, and the element of justice and higher values in the public discourse, it can be analysed how the raison d’état operates in a political doctrine. (shrink)

According to David Lewis's extreme modal realism, every waythat a world could be is a way that some concretely existingphysical world really is. But if the worlds are physicalentities, then there should be a set of all worlds, whereasI show that in fact the collection of all possible worlds is nota set. The latter conclusion remains true even outside of theLewisian framework.

We extend the theory of “Fine structure and iteration trees” to models having more than one Woodin cardinal.

We study the relations between two consequences of the Continuum Hypothesis discovered by Wacław Sierpiński, concerning uniform continuity of continuous functions and uniform convergence of sequences of real-valued functions, defined on subsets of the real line of cardinality continuum.

Assume ZF + AD +V=L and letκ< Θ be an uncountable cardinal. We show thatκis Jónsson, and that if cof = ω thenκis Rowbottom. We also establish some other partition properties.

The aim of this paper is to articulate the basic elements of a comprehensive ethic of academic administration, organized around a set of three cardinal virtues: commitment to the good of the institution; good administrative judgment; and conscientiousness in discharging the duties of the office. In addition to explaining this framework and defending its adequacy, the paper develops an account of the nature of integrity, and argues that the three cardinal virtues of academic administration can be captured in (...) the concept of integrity in academic administration. The Aristotelian basis for this framework is summarized, and its central ideas are illustrated through a variety of applications. (shrink)

Anuj Dawar poses two questions which give finitary analogies to the Löwenheim–Skolem theorems. Grohe , has shown that the first of these, which corresponds to the downward Löwenheim–Skolem theorem, has a negative answer. In this paper we combine Grohe's technique with that of Robinson's famous paper ) to show that the second question, which corresponds to the upward Löwenheim–Skolem theorem, also has a negative answer.

The notions of finite and infinite second-order characterizability of cardinal and ordinal numbers are developed. Several known results for the case of finite characterizability are extended to infinite characterizability, and investigations of the second-order theory of ordinals lead to some observations about the Fraenkel-Carnap question for well-orders and about the relationship between ordinal characterizability and ordinal arithmetic. The broader significance of cardinal characterizability and the relationships between different notions of characterizability are also discussed.

In lieu of an abstract, here is a brief excerpt of the content:Picking Up the Pieces of a Shattered Culture:Abandoning Sartre for AquinasR. E. HouserI expect to die in my bed, my successor will die in prison, and his successor will die a martyr in the public square. Then his successor will pick up the shards of a ruined society and slowly help rebuild civilization, as the Church has done so often in human history.—Francis Cardinal George (2010)Here I propose (...) to present a feature of the ethics of Thomas Aquinas, his ethical doctrine of virtue, concentrating on the four cardinal virtues—justice, prudence or good practical judgment, courage, and temperance. Plato first selected them as the four premier virtues, but they go back at least to Homer, for whom wily Odysseus exemplified prudence, warlike Achilles courage, old Nestor justice, Penelope and Telemachus, Odysseus' wife and son, temperance. And I offer them as central to living the morally good life required to "rebuild civilization" in an America that so fair and judicious a man as Cardinal George could call a "ruined society." To see the need for Thomas's solution, we must begin with the problem. Now, all societies depend upon the deep and fundamental ideas of the most important thinkers, prophets and sages, saints, and yes, even philosophers. Their ideas "trickle down" to influence regular folk, like you and me, those who whose hard work actually makes a society thrive, or die. And when those fundamental ideas change, the moral principles of those who embrace those ideas change, and consequently their actions change, and as a result the wider society in which they live changes, for better or worse. Such effects can now be felt across America [End Page 135] by all; but most of us are unaware of the primary intellectual agents of these changes, since they operate at the level of fundamental ideas, not politics or business or day-to-day life. And often their originators have died before the ideas reach full cultural force. So, ignorance of history makes it difficult to deal with present problems caused by those ideas.My first conclusion will be that a major cause of the current coarsening of our culture is the group of ideas of the French philosopher Jean Paul Sartre, even though most Americans have never heard of him, and most of those who recognize his name have not read his works carefully enough to see the connection. Why Sartre? In the honors program at my university, I had a bright young lady of Muslim heritage who had gone to a lycée, one of France's elite classical secondary schools. "Did you read the classics?" "Yes." Expecting Montaigne or Pascal, I asked, "Who?" Her answer: "Sartre." So let us begin with the theory and practice of Sartre and his long-time lover Simone Beauvoir, whose book The Second Sex was an intellectual spark for the women's liberation movement. Then we turn to some real-life examples of Sartrean influence on ourselves and American culture. Finally, for the best remedy for non-Christian and Christian alike, I recommend Aquinas's doctrine of virtue, when taken in its full scope.Sartre's Ethics of Absolute FreedomIn 1946, the year I was born, Sartre gave a lecture to a meeting of French librarians, which became his most popular book, Existentialism is a Humanism. There he laid out the core of his thought about God, humans, and their ethical behavior. Here I arrange his main ideas in an order that facilitates comparison with Aquinas, beginning with his view of human ontology and proceeding to its impact on the four cardinal virtues.Human OntologySartre turned his back on the traditional metaphysical view that every creature is an individual substance defined by a nature or essence it has in common with all other things of that type. In his "first principle of existentialism," Sartre skips over other animate things, plants and animals, and compares humans to inanimate objects:Let us consider... a book or a paper-cutter: here is an object made by an artisan whose inspiration came from a concept.... One cannot postulate a man who makes a paper-cutter but does not know what it is... (shrink)

This paper has two goals: 1) to understand justice as a cardinal virtue, according to Aquinas; and 2) to use his conception of justice as a cardinal virtue to understand how one engages in acts of “general” justice. The argument proceeds in four stages: 1) how Aquinas understands the virtues by looking to their “objects”; 2) the two distinct “modes” of the four cardinal virtues, as “general” and “specific” virtues; 3) the triangle of three kinds of justice, (...) seen in terms of their “objects”; 4) Aquinas’s doctrine of justice as a “general” virtue (ST 2-2.58.5–6) shows that we can perform operations of “general” justice in two ways, as do the ruler and his minsters, and as ordinary folk do. Surprisingly, it is the latter mode of acting for “general” justice that is primary, not the former. (shrink)

Given a structure for a first-order language L, two objects of its domain can be indiscernible relative to the properties expressible in L, without using the equality symbol, and without actually being the same. It is this relation that interests us in this paper. It is called Leibniz equality. In the paper we study systematically the problem of its definibility mainly for classes of structures that are the models of some equality-free universal Horn class in an infinitary language Lκκ, where (...) κ is an infinite regular cardinal. (shrink)

“Small” large cardinal notions in the language of ZFC are those large cardinal notions that are consistent with V = L. Besides their original formulation in classical set theory, we have a variety of analogue notions in systems of admissible set theory, admissible recursion theory, constructive set theory, constructive type theory, explicit mathematics and recursive ordinal notations (as used in proof theory). On the face of it, it is surprising that such distinctively set-theoretical notions have analogues in such (...) disaparate and relatively constructive contexts. There must be an underlying reason why that is possible (and, incidentally, why “large” large cardinal notions have not led to comparable analogues). My long term aim is to develop a common language in which such notions can be expressed and can be interpreted both in their original classical form and in their analogue form in each of these special constructive and semi-constructive cases. This is a program in progress. What is done here, to begin with, is to show how that can be done to a considerable extent in the settings of classical and admissible set theory (and thence, admissible recursion theory). The approach taken here is to expand the language of set theory to allow us to talk about (possibly partial) operations applicable both to sets and to operations and to formulate the large cardinal notions in question in terms of operational closure conditions; at the same time only minimal existence axioms are posited for sets. The resulting system, called Operational Set Theory, is a partial adaptation to the set-theoretical framework of the explicit mathematics framework Feferman (1975). The.. (shrink)

Ranking finite subsets of a given set X of elements is the formal object of analysis in this article. This problem has found a wide range of economic interpretations in the literature. The focus of the article is on the family of rankings that are additively representable. Existing characterizations are too complex and hard to grasp in decisional contexts. Furthermore, Fishburn (1996), Journal of Mathematical Psychology 40, 64–77 showed that the number of sufficient and necessary conditions that are needed to (...) characterize such a family has no upper bound as the cardinality of X increases. In turn, this article proposes a way to overcome these difficulties and allows for the characterization of a meaningful (sub)family of additively representable rankings of sets by means of a few simple axioms. Pattanaik and Xu’s (1990), Recherches Economiques de Louvain 56, 383–390) characterization of the cardinality-based rule will be derived from our main result, and other new rules that stem from our general proposal are discussed and characterized in even simpler terms. In particular, we analyze restricted-cardinality based rules, where the set of “focal” elements is not given ex-ante; but brought out by the axioms. (shrink)

8.3 The consistency proof -- 8.4 Applications of the consistency proof -- 8.5 Second-order arithmetic -- Problems -- Chapter 9: Set Theory -- 9.1 Axioms for sets -- 9.2 Development of set theory -- 9.3 Ordinals -- 9.4 Cardinals -- 9.5 Interpretations of set theory -- 9.6 Constructible sets -- 9.7 The axiom of constructibility -- 9.8 Forcing -- 9.9 The independence proofs -- 9.10 Large cardinals -- Problems -- Appendix The Word Problem -- Index.

A Transition to Proof: An Introduction to Advanced Mathematics describes writing proofs as a creative process. There is a lot that goes into creating a mathematical proof before writing it. Ample discussion of how to figure out the "nuts and bolts'" of the proof takes place: thought processes, scratch work and ways to attack problems. Readers will learn not just how to write mathematics but also how to do mathematics. They will then learn to communicate mathematics effectively. The text emphasizes (...) the creativity, intuition, and correct mathematical exposition as it prepares students for courses beyond the calculus sequence. The author urges readers to work to define their mathematical voices. This is done with style tips and strict "mathematical do’s and don’ts", which are presented in eye-catching "text-boxes" throughout the text. The end result enables readers to fully understand the fundamentals of proof. Features: The text is aimed at transition courses preparing students to take analysis Promotes creativity, intuition, and accuracy in exposition The language of proof is established in the first two chapters, which cover logic and set theory Includes chapters on cardinality and introductory topology. (shrink)

The main result of this paper is that if $\kappa$ is not a weakly Mahlo cardinal, then the following two conditions are equivalent: 1. $\mathscr{P}(\kappa)/ \mathscr{J}$ is $\kappa^+$-complete. 2. $\mathscr{J}$ is a prenormal ideal. Our result is a generalization of an announcement made in [Z]. We say that $\mathscr{J}$ is selective iff for every $\mathscr{J}$-function $f: \kappa \rightarrow \kappa$ there is a set $X \in \mathscr{J}$ such that $f\mid(\kappa - X)$ is one-to-one. Our theorem provides a positive partial answer (...) to a question of B. Weglorz from [BTW, p. 90], viz.: is every selective ideal $\mathscr{J}$, with $\mathscr{P}(\kappa)/\mathscr{J} \kappa^+$-complete, isomorphic to a normal ideal? The theorem is also true for fine ideals on $\lbrack\lambda\rbrack^{<\kappa}$ for any $\kappa \leq \lambda$, i.e. if $\kappa$ is not a weakly Mahlo cardinal then the Boolean algebra $\mathscr{P} (\lbrack\lambda\rbrack^{<\kappa})/\mathscr{J}$ is $\lambda^+$-complete iff $\mathscr{J}$ is a prenormal ideal (in the sense of $\lbrack\lambda\rbrack^{<\kappa})$. (shrink)

The main result of this paper is that if κ is not a weakly Mahlo cardinal, then the following two conditions are equivalent: 1. P(κ)/ J is κ+-complete. 2. J is a prenormal ideal. Our result is a generalization of an announcement made in [Z]. We say that J is selective iff for every J-function f: κ → κ there is a set X ∈ J such that f∣(κ - X) is one-to-one. Our theorem provides a positive partial answer (...) to a question of B. Weglorz from [BTW, p. 90], viz.: is every selective ideal J, with P(κ)/J κ+-complete, isomorphic to a normal ideal? The theorem is also true for fine ideals on $\lbrack\lambda\rbrack^{<\kappa}$ for any κ ≤ λ, i.e. if κ is not a weakly Mahlo cardinal then the Boolean algebra $\mathscr{P} (\lbrack\lambda\rbrack^{<\kappa})/\mathscr{J}$ is λ+-complete iff J is a prenormal ideal (in the sense of $\lbrack\lambda\rbrack^{<\kappa})$. (shrink)

Saunders Mac Lane has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system of set theory of which the axioms are Extensionality, Null Set, Pairing, Union, Infinity, Power Set, Restricted Separation, Foundation, and Choice, to which system, afforced by the principle, , of Transitive Containment, we shall refer as . His system is naturally related to systems derived from topos-theoretic notions concerning the category of sets, and is, as Mac Lane emphasises, one that is (...) adequate for much of mathematics. In this paper we show that the consistency strength of Mac Lane's system is not increased by adding the axioms of Kripke–Platek set theory and even the Axiom of Constructibility to Mac Lane's axioms; our method requires a close study of Axiom H, which was proposed by Mitchell; we digress to apply these methods to subsystems of Zermelo set theory , and obtain an apparently new proof that is not finitely axiomatisable; we study Friedman's strengthening of , and the Forster–Kaye subsystem of , and use forcing over ill-founded models and forcing to establish independence results concerning and ; we show, again using ill-founded models, that proves the consistency of ; turning to systems that are type-theoretic in spirit or in fact, we show by arguments of Coret and Boffa that proves a weak form of Stratified Collection, and that is a conservative extension of for stratified sentences, from which we deduce that proves a strong stratified version of ; we analyse the known equiconsistency of with the simple theory of types and give Lake's proof that an instance of Mathematical Induction is unprovable in Mac Lane's system; we study a simple set theoretic assertion—namely that there exists an infinite set of infinite sets, no two of which have the same cardinal—and use it to establish the failure of the full schema of Stratified Collection in ; and we determine the point of failure of various other schemata in . The paper closes with some philosophical remarks. (shrink)

This is a survey paper giving a self-contained account of Shelah's theory of the pcf function pcf={cf:D is an ultrafilter on a}, where a is a set of regular cardinals such that acardinal arithmetic, the existence of Jonsson algebras, and partition calculus.

Axiomatic set theory is the concern of this book. More particularly, the authors prove results about the coding of models M, of Zermelo-Fraenkel set theory together with the Generalized Continuum Hypothesis by using a class 'forcing' construction. By this method they extend M to another model L[a] with the same properties. L[a] is Gödels universe of 'constructible' sets L, together with a set of integers a which code all the cardinality and cofinality structure of M. Some applications are also considered. (...) Graduate students and research workers in set theory and logic will be especially interested by this account. (shrink)

Marcus Giaquinto claims that finite cardinal numbers are sensible properties, and that the smallest ones are known by acquaintance. In this paper I compare Giaquinto’s epistemology to the Russellian one with which it invites comparison, before showing how it is subject to a version of Jody Azzouni’s “epistemic role” objection. Then I argue that the source of this problem is Giaquinto’s misconception that numbers, like quantities, are sensible properties. Finally, I offer a sketch of a theory of how we (...) grasp finite cardinals on the assumption that they are not sensible, and show why this is not subject to the epistemic role objection. (shrink)