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)

The March 2002 symposium Human Dignity and Reproductive Technology brought together philosophers, theologians, scientists, lawyers, and scholars from across the United States. The essays of this book are the contributions of the symposium's participants.

The interaction between large cardinals, determinacy of two-person perfect information games, and inner model theory has been a singularly powerful driving force in modern set theory during the last three decades. For the outsider the intellectual excitement is often tempered by the somewhat daunting technicalities, and the seeming length of study needed to understand the flow of ideas. The purpose of this article is to try and give a short, albeit rather rough, guide to the broad lines of development.

Let κ be a cardinal, and let H κ be the class of sets of hereditary cardinality less than κ ; let τ (κ) > κ be the height of the smallest transitive admissible set containing every element of {κ}∪H κ . We show that a ZFC-definable notion of long unfoldability, a generalisation of weak compactness, implies in the core model K, that the mouse order restricted to H κ is as long as τ. (It is known that some weak (...) large cardinal property is necessary for the latter to hold.) In other terms we delimit its strength as follows: Theorem Con(ZFC+ω2-Π 1 1-Determinacy) ⇒ ⇒Con(ZFC+V=K+∃ a long unfoldable cardinal ⇒ ⇒Con(ZFC+∀X(X # exists) + ‘‘ $\forall D \subseteq \omega_1 D$ is universally Baire ⇔ ∃r∈R(D∈L(r)))’’, and this is set-generically absolute). We isolate a notion of ω-closed cardinal which is weaker than an ω1-Erd\ os cardinal, and show that this bounds the first long unfoldable: Theorem Let κ be ω -closed. Then there is a long unfoldable ł<κ. (shrink)

Let P be any one of the following combinatorial properties: weak P-pointness, weak (semi-) Q-pointness, weak (semi-)selectivity, ω-closedness. We deal with the following two questions: (1) What is the least cardinal κ such that there exists an ideal with κ many generators that does not have the property P? (2) Can one extend every ideal with the property P to a prime ideal with the property P?

P. T. Geach, notoriously, holds the Relative Identity Thesis, according to which a meaningful judgment of identity is always, implicitly or explicitly, relative to some general term. ‘The same’ is a fragmentary expression, and has no significance unless we say or mean ‘the same X’, where ‘X’ represents a general term (what Frege calls a Begriffswort or Begriffsausdruck). (P. T. Geach, Mental Acts (London: Routledge and Kegan Paul, 1957), p. 69. I maintain that it makes no sense to judge whether (...) things are ‘the same’, or remain ‘the same’, unless we add or understand some general term - ‘the same F’. (P. T. Geach, Reference and Generality, third Edition (Ithaca, N.Y.: Cornell University Press, 1980), pp. 63f. I am arguing for the thesis that identity is relative. When one says ‘x is identical with y’, this, I hold, is an incomplete expression; it is short for ‘x is the same A as y’, where ‘A’ represents some count noun understood from the context of utterance - or else, it is just a vague expression of a half-formed thought. (P. T. Geach, ‘Identity,’ Review of Metaphysics 21 (1967-8), p. 3.) One of the ways Geach seeks to support this is by tying it to the well nigh universally admired Fregean thesis about cardinality. (shrink)

We give limits defined in terms of abstract pointclasses of the amount of determinacy available in certain canonical inner models involving strong cardinals. We show for example: Theorem A. $\mathrm{D}\mathrm{e}\mathrm{t}\text{\hspace{0.17em}}({\mathrm{\Pi }}_{1}^{1}-\mathrm{I}\mathrm{N}\mathrm{D})$ ⇒ there exists an inner model with a strong cardinal. Theorem B. Det(AQI) ⇒ there exist type-1 mice and hence inner models with proper classes of strong cardinals. where ${\mathrm{\Pi }}_{1}^{1}-\mathrm{I}\mathrm{N}\mathrm{D}\phantom{\rule{0ex}{0ex}}$ (AQI) is the pointclass of boldface ${\mathrm{\Pi }}_{1}^{1}$ -inductive (respectively arithmetically quasi-inductive) sets of reals.

We show that, like singular cardinals, and weakly compact cardinals, Jensen's core model K for measures of order zero [4] calculates correctly the successors of Jónsson cardinals, assuming $O^{Sword}$ does not exist. Namely, if $\kappa$ is a Jónsson cardinal then $\kappa^+ = \kappa^{+K}$ , provided that there is no non-trivial elementary embedding $j:K \longrightarrow K$ . There are a number of related results in ZFC concerning $\cal{P}(\kappa)$ in V and inner models, for $\kappa$ a Jónsson or singular cardinal.

• We define a notion of order of indiscernibility type of a structure by analogy with Mitchell order on measures; we use this to define a hierarchy of strong axioms of infinity defined through normal filters, the α-weakly Erdős hierarchy. The filters in this hierarchy can be seen to be generated by sets of ordinals where these indiscernibility orders on structures dominate the canonical functions.• The limit axiom of this is that of greatly Erdős and we use it to calibrate (...) some strengthenings of the Chang property, one of which, CC+, is equiconsistent with a Ramsey cardinal, and implies that where K is the core model built with non-overlapping extenders — if it is rigid, and others which are a little weaker. As one corollary we have:TheoremIf then there is an inner model with a strong cardinal. • We define an α-Jónsson hierarchy to parallel the α-Ramsey hierarchy, and show that κ being α-Jónsson implies that it is α-Ramsey in the core model. (shrink)

There is no prospect of discovering measurable cardinals by radio astronomy, but this does not mean that higher set theory is entirely irrelevant to applied mathematics broadly construed. By way of example, the bearing of some celebrated descriptive-set-theoretic consequences of large cardinals on measurable-selection theory, a body of results originating with a key lemma in von Neumann’s work on the mathematical foundations of quantum theory, and further developed in connection with problems of mathematical economics, will be considered from a philosophical (...) point of view. (shrink)

1. The Tractatus doctrine of saying and showing In a letter to Russell dated 19.4.1919, written shortly after he had finished the Tractatus, Wittgenstein told Russell that the main contention of the book, to which all else, including the account of logic, is subsidiary, ‘is the theory of what can be expressed (gesagt) by prop[osition]s -- i.e. by language -- (and, which comes to the same, what can be thought) and what cannot be expressed by prop[osition]s, but only shown (gezeigt); (...) which I believe is the cardinal problem of philosophy’ (CL 68). This was emphasized in both preface and conclusion of the book. The preface observes that the whole sense of the book can be summed up in the following words: ‘what can be said at all can be said clearly, and what we cannot talk about we must pass over in silence’ (TLP p.3). The conclusion of the book (TLP 7) simply repeats this. The preceding three remarks, however, are noteworthy. They make three claims. First, ‘There are, indeed, things that cannot be put into words. They make themselves manifest. They are what is mystical’ (TLP 6.522). This reiterates the leitmotif of the book, namely that there are things that cannot, by the very nature of representation, be said. But though they cannot be said, they are shown by features of the relevant system of representation. Secondly, the correct method in philosophy would really be to say nothing except what can be said, i.e. empirical propositions that have nothing to do with philosophy, and then, when someone wanted to say something metaphysical, to demonstrate to him that he had failed to give meaning to certain signs in his propositions (TLP 6.53). This method, of course, is not the method that has been followed throughout the book, which consists almost exclusively of modal assertions concerning what must, can or cannot be thus and so in reality, in language and in the relation between language and reality.. (shrink)

Ever since the first edition appeared in 1952, Wilder's book has been a mainstay of courses in the philosophy and foundations of mathematics, and deservedly so, for it covers most of the topics which provide an insight into the nature of this formal science. There are two parts: the first is a rapid but thorough survey of the axiomatic method, set theory, especially infinite sets, cardinal and ordinal numbers, the linear continuum, and the theory of groups with reference to foundational (...) problems. The second part develops various views on the nature of the foundations of mathematics: it begins with a look at the naïve 19th-century approach and its transformation into Zermelo's axiomatic set theory; the Logicistic thesis is covered in great detail; intuitionism and Hilbert's formalistic view come next; the last chapter discusses the cultural setting of mathematics. The changes since the first edition have been in the way of making certain presentations more relevant to current problems. There is an excellent bibliography. This book could be used in a one semester course on philosophy or on foundations of mathematics, or in a longer course in which symbolic logic was developed extensively.—P. J. M. (shrink)

Reflection Principles are commonly thought to produce only strong axioms of infinity consistent with V = L. It would be desirable to have some notion of strong reflection to remedy this, and we have proposed Global Reflection Principles based on a somewhat Cantorian view of the universe. Such principles justify the kind of cardinals needed for, inter alia , Woodin’s Ω-Logic.

A succinct introduction to mathematical logic and set theory, which together form the foundations for the rigorous development of mathematics. Suitable for all introductory mathematics undergraduates, Notes on Logic and Set Theory covers the basic concepts of logic: first-order logic, consistency, and the completeness theorem, before introducing the reader to the fundamentals of axiomatic set theory. Successive chapters examine the recursive functions, the axiom of choice, ordinal and cardinal arithmetic, and the incompleteness theorems. Dr. Johnstone has included numerous exercises designed (...) to illustrate the key elements of the theory and to provide applications of basic logical concepts to other areas of mathematics. (shrink)

A small large cardinal upper bound in V for proving when certain subsets of ω 1 (including the universally Baire subsets) are precisely those constructible from a real is given. In the core model we find an exact equivalence in terms of the length of the mouse order; we show that $\forall B \subseteq \omega_1 \lbrack B$ is universally Baire $\Leftrightarrow B \in L\lbrack r \rbrack$ for some real r] is preserved under set-sized forcing extensions if and only if there (...) are arbitrarily large "admissibly measurable" cardinals. (shrink)

A small large cardinal upper bound in $V$ for proving when certain subsets of $\omega_1$ are precisely those constructible from a real is given. In the core model we find an exact equivalence in terms of the length of the mouse order; we show that $\forall B \subseteq \omega_1 \lbrack B$ is universally Baire $\Leftrightarrow B \in L\lbrack r \rbrack$ for some real $r\rbrack$ is preserved under set-sized forcing extensions if and only if there are arbitrarily large "admissibly measurable" cardinals.

We discuss some of the relationships between the notion of "mutual stationarity" of Foreman and Magidor and measurability in inner models. The general thrust of these is that very general mutual stationarity properties on small cardinals, such as the ℵns, is a large cardinal property. A number of open problems, theorems, and conjectures are stated.

A new axiomatization of set theory, to be called Bernays-Boolos set theory, is introduced. Its background logic is the plural logic of Boolos, and its only positive set-theoretic existence axiom is a reflection principle of Bernays. It is a very simple system of axioms sufficient to obtain the usual axioms of ZFC, plus some large cardinals, and to reduce every question of plural logic to a question of set theory.

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

The familiar Vendler-Kenny scheme of verb-types, viz., performances (further differentiated by Vedler into accomplishments and achievements), activities, and states, is too narrow in two important respects. First, it is narrow linguistically. It fails to take into account the phenomenon of verb aspect. The trichotomy is not one of verbs as lexical types but of predications. Second, the trichotomy is narrow ontologically. It is a specification in the context of human agency of the more fundamental, topic-neutral trichotomy, event-process-state.The central component in (...) this ontological trichotomy, event, can be sharply differentiated from its two flanking components by adapting a suggestion by Geoffrey N. Leech and others that the contrast between perfective and imperfective aspect in verbs corresponds to the count/mass distinction in the domain of nouns. With the help of two distinctions, of cardinal count adverbials versus frequency adverbials, and of occurrence versus associated occasion, two interrelated criteria for event predication are developed. Accordingly, Mary capsized the boat is an event predication because (a) it is equivalent to There was at least one capsizing of the boat by Mary, or (b) because it admits cardinal count adverbials, e.g., at least once, twice, three times. Ontologically speaking, events are defined as those occurrences that are inherently countable. (shrink)

The great logician Gottlob Frege attempted to provide a purely logical foundation for mathematics. His system collapsed when Bertrand Russell discovered a contradiction in it. Thereafter, mathematicians and logicians, beginning with Russell himself, turned in other directions to look for a framework for modern abstract mathematics. Over the past couple of decades, however, logicians and philosophers have discovered that much more is salvageable from the rubble of Frege's system than had previously been assumed. A variety of repaired systems have been (...) proposed, each a consistent theory permitting the development of a significant portion of mathematics. This book surveys the assortment of methods put forth for fixing Frege's system, in an attempt to determine just how much of mathematics can be reconstructed in each. John Burgess considers every proposed fix, each with its distinctive philosophical advantages and drawbacks. These systems range from those barely able to reconstruct the rudiments of arithmetic to those that go well beyond the generally accepted axioms of set theory into the speculative realm of large cardinals. For the most part, Burgess finds that attempts to fix Frege do less than advertised to revive his system. This book will be the benchmark against which future analyses of the revival of Frege will be measured. (shrink)

We consider the structure of all labeled trees, called also infinite terms, in the first order language with function symbols in a recursive signature of cardinality at least two and at least a symbol of arity two, with equality and a binary relation symbol which is interpreted to be the subtree relation. The existential theory over of this structure is decidable (see Tulipani [9]), but more complex fragments of the theory are undecidable. We prove that the theory of the structure (...) is in , where formulas are those in prenex form consisting of a string of unbounded existential quantifiers followed by a string of arbitrary quantifiers all bounded with respect to . Since the fragment of the theory was already known to be -hard (see Marongiu and Tulipani [5]), it is now established to be -complete. (shrink)

We discuss saturating ultrafilters on N, relating them to other types of nonprincipal ultrafilter. (a) There is an (ω,c)-saturating ultrafilter on $\mathbf{N} \operatorname{iff} 2^\lambda \leq \mathfrak{c}$ for every $\lambda and there is no cover of R by fewer than c nowhere dense sets. (b) Assume Martin's axiom. Then, for any cardinal κ, a nonprincipal ultrafilter on N is (ω,κ)-saturating iff it is almost κ-good. In particular, (i) p(κ)-point ultrafilters are (ω,κ)-saturating, and (ii) the set of (ω,κ)-saturating ultrafilters is invariant under (...) homeomorphisms of $\beta\mathbf{N\backslash N}$ . (c) It is relatively consistent with ZFC to suppose that there is a Ramsey p(c)-point ultrafilter which is not (ω,c)-saturating. (shrink)

Let Mn♯ denote the minimal active iterable extender model which has n Woodin cardinals and contains all reals, if it exists, in which case we denote by Mn the class-sized model obtained by iterating the topmost measure of Mn class-many times. We characterize the sets of reals which are Σ1-definable from R over Mn, under the assumption that projective games on reals are determined:1. for even n, Σ1Mn=⅁RΠn+11;2. for odd n, Σ1Mn=⅁RΣn+11.This generalizes a theorem of Martin and Steel for L, (...) that is, the case n=0. As consequences of the proof, we see that determinacy of all projective games with moves in R is equivalent to the statement that Mn♯ exists for all n∈N, and that determinacy of all projective games of length ω2 with moves in N is equivalent to the statement that Mn♯ exists and satisfies AD for all n∈N. (shrink)

This is not a textbook in mathematical physics—excepting for one chapter one need not possess much more than geometry and elementary algebra—rather it is a philosophically reflective examination of the cardinal features of special relativity theory. Throughout the book Bohm is not merely doing physics, but thinking about doing physics as well. This metatheoretical reflexion appears in chapters concerning pre-Einsteinian notions of relativity, attempts to save the aether theories, the "ambiguity" of space-time measurements in the new cosmology, "common sense" notions (...) of space and time, and the falsification of scientific theories. There is a long appendix dealing with physics and perception—the relation between scientific objects and perceptual processes. One will, however, by working through the text learn plenty of physics in a rigorous and concise fashion. The author wisely does not attempt to cover the mathematically far more difficult and philosophically more profound General Theory of Einstein. In the area chosen, Bohm has written clearly and felicitously; this should serve a model for others who like to take their physics with a dollop of philosophy.—P. J. M. (shrink)

This is a text for a one or two semester course on axiomatic set theory; the goal is to introduce and develop one system of set theory in a complete and thorough way, presupposing only the elusive "mathematical maturity" of the reader. There are nine chapters which begin with a development of propositional and predicate logic oriented toward set theory and develop the Zermelo-Fraenkel system in exceptional detail. The book starts slowly, the first 120 pages being devoted to logical preliminaries (...) and the introduction of axioms; it gathers speed, however, and the remaining chapters include treatment of the algebra of classes, relations and functions, order of various sorts and Zorn's lemma, real numbers and equivalence classes, the equipollence of sets, similarity as a relation of sets, the ordinal numbers and their arithmetic, the cardinal numbers and their arithmetic. These later chapters are extremely detailed and comprehensive, presenting a wealth of material. There are numerous exercises, many quite difficult, at the end of each chapter, along with a summary of that chapter's content. Because of the single-mindedness of the book, there is little reference to other systems of set theory, but this is not a drawback at all. The presentation is orthodox, but not dull; careful, but not generally pedantic or repetitive. This makes it a good text for self-study.—P. J. M. (shrink)

We consider the following question of Kunen: Does Con(ZFC + ∃M a transitive inner model and a non-trivial elementary embedding j: M $\longrightarrow$ V) imply Con (ZFC + ∃ a measurable cardinal)? We use core model theory to investigate consequences of the existence of such a j: M → V. We prove, amongst other things, the existence of such an embedding implies that the core model K is a model of "there exists a proper class of almost Ramsey cardinals". Conversely, (...) if On is Ramsey, then such a j, M are definable. We construe this as a negative answer to the question above. We consider further the consequences of strengthening the closure assumption on j to having various classes of fixed points. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Paul Oskar Kristeller 1905–1999Edward P. MahoneyPaul Oskar Kristeller was without doubt one of the most productive and accomplished scholars of this century. He received an excellent education in the classics at the Mommsen-Gymnasium in his native Berlin before going to the University of Heidelberg in 1923. There he pursued studies in a wide range of subjects, including medieval history, German literature, physics, and art history. The philosophy professors who (...) much influenced him both at Heidelberg and at Freiburg and Marburg were Ernst Hoffman, Karl Jaspers, Richard Kroner, and Martin Heidegger. In 1929 he received his doctorate from Heidelberg for his dissertation Der Begriff der Seele in der Ethik des Plotin, which was directed by Hoffman and published at Tübingen the same year. He then went to Berlin, where he engaged in formal studies in classical philology before going to Freiburg to do postgraduate work in philosophy with Heidegger during the years 1931 to 1933. Heidegger approved a study of Marsilio Ficino as Kristeller’s Habili-tationschrift and helped him to secure a fellowship enabling him to do research on Ficino and to begin writing. After visiting Italy during this period and taking into account the changed political scene in Germany, Kristeller decided in 1934 to move to Italy permanently. He taught in Florence in a high school for German Jewish students until he secured in 1935, with the help of Giovanni Gentile, a lectureship in German at the Scuola Superiore and the University of Pisa. He maintained throughout his academic career a personal and warm attachment to the Scuola Superiore. Moreover, he enjoyed throughout his subsequent career a warm and personal friendship with various Italian scholars that was first developed during his Italian stay in the thirties.While still in Italy, Kristeller completed his book on Ficino in 1937; he prepared an Italian version the next year. However, both the German and Italian versions would be published only years later. During this period he also engaged in manuscript studies in the major Italian cities, searching in particular for Ficino manuscripts. The result of these labors was his two-volume work, Supplementum Ficinianum: Marsilii Ficini Philosophi Platonici Opuscula Inedita et Dispersa, which was published by the Scuola Normale in 1937. He considered his “teachers” in manuscript studies to be Giovanni Cardinal Mercati, Monsignor Auguste Pelzer, and Ludwig Bertalot. Giovanni Gentile aided the publication of Kristeller’s Supplementum Ficinianum and subsequently helped him to emigrate to the United States. Kristeller always maintained a loyalty to Gentile and felt gratitude to him for extending him such help.In 1938 Kristeller lost his position at Pisa because of the Fascist anti-Semitic laws, but he was permitted to go to Rome where he assisted Bertalot in manuscript research at the Vatican library. Through the intervention of Roland H. Bainton, Dean P. Lockwood, Dom Anselm Strittmater, O.S.B., and other American [End Page 758] friends Kristeller was able to leave for the United States. He arrived at New York in February 1939 and spent the Spring term at Yale University. He was given an adjunct position at Columbia University for the fall term, holding the rank of Associate until 1948, when he was made an Associate Professor of Philosophy. By that time he had established himself as a major force in the development of Renaissance studies in the United States. His book on Ficino was translated into English as The Philosophy of Marsilio Ficino and published in 1943 by the Columbia University Press. The Italian version was published at Florence in 1953; a revised edition, containing a new bibliography that had been brought up to date, appeared in 1988.Kristeller published a large number of articles on a wide range of topics, establishing himself as a major interpreter of the significance of Renaissance humanism as well as a leading authority not only on Ficino and Renaissance Platonism but also on Pietro Pomponazzi and Renaissance Aristotelianism. He developed at Columbia bonds of friendship with Ernest A. Moody and John Herman Randall, Jr., both of whom had strong interests in late-medieval Aristotelianism. With Randall and Ernst Cassirer he prepared what would become a bestseller, the Renaissance Philosophy of Man: Selections... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Mercy in Aquinas: Help from the Commentatorial TraditionRomanus Cessario O.P. and Cajetan Cuddy O.P.Omnes semitae Domini misericordia et veritas(Psalm 24:10)IN QUESTION 21, article 3 of the first part of the Summa theologiae, St. Thomas Aquinas outlines the dynamics of mercy:A person is said to be merciful [misericors], as being, so to speak, miserable at heart [miserum cor]; being affected with sorrow [tristitia] at the misery of another as though (...) it were his own. Hence it follows that he endeavors to dispel the misery of this other, as if it were his; and this is the effect of mercy.1However, when explaining the mercy of God, Aquinas carefully distinguishes the effective and the affective within mercy: “Mercy is especially to be attributed to God, as seen in its effect, but not as an affection of passion.”2 “It does most properly belong to Him to dispel that misery,” but “to sorrow, therefore, over the misery of others belongs not to God.” How then can one claim that God is merciful? Aquinas locates the formal nature of God’s mercy within his power to relieve the misery of the miserable: “in so far as perfections given to things by God expel defects, it [i.e., the communication of perfection] belongs [End Page 329] to mercy.”3 Aquinas, therefore, situates the divine mercy within the divine perfection—a perfection powerful enough to remove all defects.Contemporary theologians may find this account of the divine mercy somewhat unsatisfying. In our conception, a sharing in the misery of another stands as an essential element of authentic mercy. Hence, our untutored experience may incline us to press Aquinas and to ask, Why, exactly, does sorrow over the misery of others not belong to God? If one seeks clarification regarding this point within the abovementioned passage, one does not find it. Aquinas does not here expound at any length upon the reasons why God does not share the misery of the miserable.Thankfully we are not the first students of theology to approach the writings of Aquinas. Nor are we the first to observe the prima facie oddness of a seemingly impassive God within Aquinas’s consideration of mercy. Throughout history, theologians of profound elegance and insight have sought wisdom from the Angelic Doctor. Some have even devoted their academic lives to the task of studying, expounding, and “commenting” upon the Thomistic text. In this essay, we intend to rely upon—and thereby highlight—a few of the valuable insights into divine mercy these Thomist commentators offer within their commentaries. Because the Thomist commentatorial tradition includes far too many figures to summarize in a few pages, we have decided to draw from only a small handful of the major Thomist commentators within a limited period of theological history, roughly from the sixteenth through the eighteenth centuries.4 Moreover, while the Thomist commentators exhibit a wide range of uniqueness and nuance amongst themselves, we might observe in our analysis a fundamental first principle uniting these major figures: the real distinction between act and potency (as well as between form and matter). [End Page 330] While philosophical in nature, this key Thomistic principle serves as the hermeneutical key for understanding what drew the commentators to Aquinas and the essential unity of their theological project.I. Infinitely Powerful MercyWhy does God’s mercy lack any misery? The sixteenth-century Thomist commentator Thomas de Vio Cardinal Cajetan (1469–1534) explains that “misery” implies some defect in the commiserating subject.5 God is actus purus. His being admits to no potentiality and no passivity within himself. The seventeenth-century Carmelite commentators, the Salmanticenses, note the relevance of the act-potency distinction for understanding the divine mercy: “In God, mercy is not given through the mode of a potency, but through the mode of the highest actuality... which extends to the relieving of all misery.”6 The defect of misery requires some degree of potentiality and passivity within its subject. Therefore, God can experience no misery—or even co-misery.7The seventeenth-century French Dominican Jean-Baptiste Gonet (1616–81) quotes with approval Cardinal Cajetan’s commentary on question... (shrink)

On November 8, 1894, I read before the Cambridge Philological Society a paper in which the reading and the interpretation of this passage were discussed at length. A brief report of the paper was published in the Proceedings of the Society, Nos. 37–39, p. 16; and the cardinal correction was received into the text of the Fasti which Professor G. A. Davies published in the Corpus Poetarum Latinorum. The Proceedings of the Cambridge Philological Society are indeed now among the periodical (...) publications of which the Bibliotheca Philologica Classica takes account: but an acquaintance with their earlier numbers is the privilege of the elect, and the sole mention of the correction that I have seen is an entry in the diligent summary of Ovidian literature in the Jahresbericht vol. 109 p. 282 which runs as follows ‘VI 247 ut tangat .’ Whether the correction is ‘needless’ is not for me to say; but there is excuse for thinking that those who wish to form a judgment on its character will not deem it needless to have the evidence before them. (shrink)

We determine the consistency strength of determinacy for projective games of length ω^2. Our main theorem is that $\Pi _{n + 1}^1$-determinacy for games of length ω^2 implies the existence of a model of set theory with ω + n Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals A such that M_n(A), the canonical inner model for n Woodin cardinals constructed over A, satisfies $A = R$ and the (...) Axiom of Determinacy. Then we argue how to obtain a model with ω + n Woodin cardinal from this. We also show how the proof can be adapted to investigate the consistency strength of determinacy for games of length ω^2 with payoff in $R^R\Pi _1^1$ or with σ-projective payoff. (shrink)

Given the current concern in the Soviet Union and East Europe to emancipate public education from its Stalinist past, it is understandable that educators have called for the “humanizing” of education. Yet “humanization” is a none too clear idea and must be approached, I propose, through its opposite: dehumanization. Dehumanization, itself, can be understood as the denial of the dignity of the individual — a cardinal principle of the philosophies that comprise classical and contemporary liberal theory. This principle of the (...) dignity of the individual in turn implies duties for the liberal democratic state to treat all citizens (and for all citizens to treat each other) with equal respect as ends in themselves and equally as choosers of ends, as being, at least in principle, capable of choosing and acting on the ultimate ends in their own lives and as potential “sayers of great words and doers of great deeds” in the common public enterprise. Now in claiming that the principle of the dignity of the individual is central to an understanding of a just and humane society, we do not thereby commit ourselves to those doctrines long associated with liberalism, viz., ontological and moral individualism. The form of political, or united-reformed, liberalism argued for in this paper is clearly compatible with strong forms of community, even authoritarian types of community insofar as they do not transgress the principle of the dignity of the individual as regards informed choice. As decidedly non-perfectionist and agnostic about ultimate goods (tempered by the ban on all public forms of dehumanizing behavior and treatment), this form of political liberalism is neither individualistic nor communitarian, neither capitalist nor socialist (though the state is not indifferent to their fortunes). This rendering of the liberal democratic state sets the requirements for a deep and thorough political education that goes well beyond any familiar example today. In seeking to develop, to “humanize” individuals for a world of thought and action in the common public enterprise, political education — the premier form of education — in a liberal democratic state seeks to develop individuals who are theoretically informed and practically wise — thus the emphasis on the development of rationality in all of its forms. Moreover, despite the formal agnosticism of the state concerning ultimate human ends, a political education requires deep student exposure to and critical assessment of religion, possible ways of life, and those views of the human good that humans throughout history have found worthy of pursuit. And this even if it means that students will come to reject the principle of the dignity of the individual. (shrink)

We consider the following question of Kunen: Does Con imply Con? We use core model theory to investigate consequences of the existence of such a j : M $\rightarrow$ V. We prove, amongst other things, the existence of such an embedding implies that the core model K is a model of "there exists a proper class of almost Ramsey cardinals". Conversely, if On is Ramsey, then such a j, M are definable. We construe this as a negative answer to the (...) question above. We consider further the consequences of strengthening the closure assumption on j to having various classes of fixed points. (shrink)

A structure is locally finite if every finitely generated substructure is finite; local sentences are universal sentences all models of which are locally finite. The stretching theorem for local sentences expresses a remarkable reflection phenomenon between the finite and the infinite models of local sentences. This result in part requires strong axioms to be proved; it was studied by the second named author, in a paper of this Journal, volume 53. Here we correct and extend this paper; in particular we (...) show that the stretching theorem implies the existence of inaccessible cardinals, and has precisely the consistency strength of Mahlo cardinals of finite order. And we present a sequel due to the first named author: (i) decidability of the spectrum $\operatorname{Sp}(\varphi)$ of a local sentence φ, below ω ω ; where $\operatorname{Sp}(\varphi)$ is the set of ordinals α such that φ has a model of order type α (ii) proof that $\operatorname{beth}_\omega = \sup\{\operatorname{Sp}(\varphi): \varphi \text{local sentence with a bounded spectrum}\}$ (iii) existence of a local sentence φ such that $\operatorname{Sp}(\varphi)$ contains all infinite ordinals except the inaccessible cardinals. (shrink)