We study the preservation of selective covering properties, including classic ones introduced by Menger, Hurewicz, Rothberger, Gerlits and Nagy, and others, under products with some major families of concentrated sets of reals.Our methods include the projection method introduced by the authors in an earlier work, as well as several new methods. Some special consequences of our main results are : Every product of a concentrated space with a Hurewicz S1S1 space satisfies S1S1. On the other hand, assuming the Continuum Hypothesis, (...) for each Sierpiński set S there is a Luzin set L such that L×SL×S can be mapped onto the real line by a Borel function. Assuming Semifilter Trichotomy, every concentrated space is productively Menger and productively Rothberger. Every scale set is productively Hurewicz, productively Menger, productively Scheepers, and productively Gerlits–Nagy. Assuming d=ℵ1d=ℵ1, every productively Lindelöf space is productively Hurewicz, productively Menger, and productively Scheepers.A notorious open problem asks whether the additivity of Rothberger's property may be strictly greater than addadd, the additivity of the ideal of Lebesgue-null sets of reals. We obtain a positive answer, modulo the consistency of Semifilter Trichotomy >ℵ1.Our results improve upon and unify a number of results, established earlier by many authors. (shrink)

In this paper we ask the question: to what extent do basic set theoretic properties of Loeb measure depend on the nonstandard universe and on properties of the model of set theory in which it lies? We show that, assuming Martin's axiom and κ-saturation, the smallest cover by Loeb measure zero sets must have cardinality less than κ. In contrast to this we show that the additivity of Loeb measure cannot be greater than ω 1 . Define $\operatorname{cof}(H)$ as the (...) smallest cardinality of a family of Loeb measure zero sets which cover every other Loeb measure zero set. We show that $\operatorname{card}(\lfloor\log_2(H)\rfloor) \leq \operatorname{cof}(H) \leq \operatorname{card}(2^H)$ , where card is the external cardinality. We answer a question of Paris and Mills concerning cuts in nonstandard models of number theory. We also present a pair of nonstandard universes $M \preccurlyeq N$ and hyperfinite integer H ∈ M such that H is not enlarged by N, 2 H contains new elements, but every new subset of H has Loeb measure zero. We show that it is consistent that there exists a Sierpiński set in the reals but no Loeb-Sierpiński set in any nonstandard universe. We also show that it is consistent with the failure of the continuum hypothesis that Loeb-Sierpiński sets can exist in some nonstandard universes and even in an ultrapower of a standard universe. (shrink)

The principle of Sierpinski is the assertion that there is a family of functions $\left\{ {{\varphi _n}:{\omega _1} \to {\omega _1}|n \in \omega } \right\}$ such that for every $I \in {[{\omega _1}]^{{\omega _1}}}$ there is n ε ω such that ${\varphi _n}[I] = {\omega _1}$. We prove that this principle holds if there is a nonmeager set of size ω1 answering question of Arnold W. Miller. Combining our result with a theorem of Miller it then follows that is equivalent (...) to $non\left = {\omega _1}$. Miller also proved that the principle of Sierpinki is equivalent to the existence of a weak version of a Luzin set, we will construct a model where all of these sets are meager yet $non\left = {\omega _1}$. (shrink)

Bibliography of A. A. Fraenkel (p. ix-x)--Axiomatic set theory. Zur Frage der Unendlichkeitsschemata in der axiomatischen Mengenlehre, von P. Bernays.--On some problems involving inaccessible cardinals, by P. Erdös and A. Tarski.--Comparing the axioms of local and universal choice, by A. Lévy.--Frankel's addition to the axioms of Zermelo, by R. Mantague.--More on the axiom of extensionality, by D. Scott.--The problem of predicativity, by J. R. Shoenfield.--Mathematical logic. Grundgedanken einer typenfreien Logik, von W. Ackermann.--On the use of Hilbert's [epsilon]-operator in scientific theories, (...) by R. Carnap.--Basic verifiability in the combinatory theory of restricted generality, by H. B. Curry.--Uniqueness ordinals in constructive number classes, by H. Putnam.--On the construction of models, by A. Robinson.--Interpretation of mathematical theories in the first order predicate calculus, by T. Skolem.--The elementary character of two notions from general algebra, by R. Vaught.--Foundations of arithmetic and analysis. Axiomatic method and intuitionism, by A. Heyting.--On rank-decreasing functions, by G. Kurepa.--On non-standard models for number theory, by E. Mendelson.--Concerning the problem of axiomatizability of the field of real numbers in the weak second order logic, by A. Mostowski.--Non-standard models and independence of the induction axiom, by M. O. Rabin.--Sur les ensembles raréfiés de nombres naturels, par W. Sierpinski.--Philosophy of logic and mathematics. Remarks on the paradoxes of logic and set theory, by E. W. Beth.--Logique formalisée et raisonnement juridique, par R. Feys.--Im Umkreis der sogenannten Raumprobleme, von H. Freudenthal.--Process and existence in mathematics, by H. Wang. (shrink)

The main aim of the paper was to draw attention to Wacław Sierpiński as not only a great mathematician but also a philosopher. We undertook the attempt of reconstruction of Sierpiński’s philosophy. To aim this goal we mainly based ourselves on Sierpiński’s habilitation lecture entitled _The concept of correspondence in mathematics_. The complementation of Sierpiński’s philosophical views were conclusions from his mathematical achievements, his scheme of research on The Axiom of Choice, and his attitude to this (...) axiom. (shrink)

_Imagination_ is an outstanding contribution to a notoriously elusive and confusing subject. It skillfully interrelates problems in philosophy, the history of ideas and literary theory and criticism, tracing the evolution of the concept of imagination from Hume and Kant in the eighteenth century to Ryle, Sartre and Wittgenstein in the twentieth. She strongly belies that the cultivation of imagination should be the chief aim of education and one of her objectives in writing the book has been to put forward reasons (...) why this is so. Purely philosophical treatment of the concept is shown to be related to its use in the work of Coleridge and Wordsworth, who she considers to be the creators of a new kind of awareness with more than literary implications. The purpose of her historical account is to suggest that the role of imagination in our perception and thought is more pervasive than may at first sight appear, and that the thread she traces is an important link joining apparently different areas of our experience. She argues that imagination is an essential element in both our awareness of the world and our attaching of value to it. (shrink)

Historians of technology have provided important accounts of technological innovation, but they rarely employ concepts which permit a rigorous analysis ofinvention as a mental or cognitive process. This article seeks to address this theoretical lacuna by using concepts adapted from cognitive psychology to compare the mental processes of two telephone inventors, Alexander Graham Bell and Thomas Edison. Specifically, we suggest that invention may be seen as a process in which inventors combine ideas with objects, or what we call mental models (...) and mechanical representations. The strategies by which inventors generate and manipulate these mental models and mechanical representations are what we refer to as heuristics. Using these concepts to narrate the development of the telephone, this article shows how invention can be interpreted as being much more than simply a mysterious act of individual genius. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Transformed Lives: Making Sense of Atonement Today by Cynthia S. W. CrysdaleVirginia W. LandgrafTransformed Lives: Making Sense of Atonement Today Cynthia S. W. Crysdale new york: seabury books, 2016. 192 pp. $16.00Cynthia Crysdale aims to show how atonement can have meaning for modern and postmodern Christians who reject the idea that God wills Jesus's violent death. She starts with stories of people who were estranged from God but (...) have been given grace to love God anew. Their subsequent lives are not perfect and involve multiple deaths of the old self and resurrections into the new. Yet there is a reconciliation not present beforehand: not a change in God, who has always loved us, but a change in us. Crysdale seeks to navigate between two undesirable positions: a penal substitutionary doctrine of the atonement, which she thinks posits a violent God, or a rejection of the idea of atonement altogether. She sees atonement as not only moral influence showing an example but a relationship that involves our whole being.Crysdale then presents a historical overview to show that this understanding of atonement is a plausible interpretation of scripture and tradition. Biblical sacrifices were not intended to propitiate an angry deity but to expiate human sin. New Testament writers who saw Jesus's death on the cross in continuity with these sacrifices were not implying that God was punishing Jesus. Jesus's statements about giving his life as a ransom prophesied the subversion of patron–client power relations rather than predicting a transaction whereby Jesus would pay for sin. Subsequent writers who saw Jesus as gaining victory over the devil saw God and Jesus as united against the powers and Jesus as attaining justice by love. Even Anselm, sometimes blamed for the penal substitutionary view, intended to avoid the idea of the crucifixion as God's punishment of Jesus. In the honor code of the time, either satisfaction or punishment was required. Therefore, Jesus's act of making satisfaction is precisely not punishment. Crysdale sees subsequent conflations of satisfaction and punishment as distorting Anselm's original intent and implying that God requires violent punishment of sin. [End Page 208]Crysdale then takes readers on a tour of changes in scientific, historical, and epistemological consciousness in the modern era to ask how theology might be done today. Drawing on Bernard Lonergan, she describes multiple dimensions of conversion that are all instances of grace when they occur. She then interprets the incarnation as indicating God's offer of friendship with humanity and Jesus's death and resurrection as indicating the transformation of power and oppression. Believing that an abstract atonement over the heads of human beings does not exist, she concludes with further stories of lives transformed by an encounter with God, including Martin Luther King Jr. and Dorothy Day.This book is suitable for educated lay adults, advanced undergraduates, or introductory seminary courses. Protestant audiences may find it typically Catholic, even though Crysdale is an Anglican, because it emphasizes changed lives rather than God's act of justification prior to any change in us. Theologically, it does not deal with every possible objection to its thesis. Those concerned with cultures of impunity, where malevolent lawlessness gets its way, may think that rejecting any sense of punishment of sin in the crucifixion throws the baby out with the bathwater. Recently, Fleming Rutledge (in The Crucifixion), drawing on Karl Barth, has made a point of retaining punishment as one meaning of the atonement even while emphasizing that God's wrath is always in the service of God's love. Transformed Lives does not have the last word on the atonement and would not claim to, since Crysdale wants to invite readers to do theology befitting their changing context. However, readers at many levels will benefit from its accessible treatment of its topic and invitation to reflection that befits a loving, reconciling, evil-opposing God.Virginia W. LandgrafAmerican Theological Library AssociationCopyright © 2018 Society of Christian Ethics... (shrink)

Two recent papers by Michael Burke bearing upon the persistence of people and commonplace things illustrate the fact that the quest for synchronic ontological economy is likely to encourage a disturbing diachronic proliferation of entities. This discussion argues that Burke's promise of ontological economy is seriously compromised by the fact that his proposed metaphysic does violence to standard intuitions concerning the persistence of people and commonplace things. In effect, Burke would have us achieve synchronic economy (rejection of coincident entities) by (...) postulating strongly counterintuitive transtemporal claims of numerical diversity. The argument is made that the price of Burkean economy is too high. (shrink)

This is the second of five chapters on good, and starts by making the point that it is around the question of the intrinsically good that the chief controversies about the nature of goodness or value revolve, for most theories of value may be divided into those that treat it as a quality and those that treat it as a relation between that which has value and something else ; Ross says that it seems clear that any view that treats (...) goodness as a relation between that which is good and something else denies that anything is intrinsically good. The chapter first discusses R. B. Perry's relational view of value—in the sense of good, and other theories that also identify goodness with some relation, either as a relation between that which is good and some or all of its elements, or as a relation between some or all of its elements, or as a relation between it or some or all of its elements and something else. Next, the psychological theories of good are discussed; these, as a rule, hold that a thing being good means either that some person or persons have some feelings towards it, or some person or persons think it to be good; this is an objective view of good. The rest of the chapter looks at both R. B. Perry's and G. E. Moore's arguments on these theories, at Ross's own views, at badness as incompatible with goodness or as an element of a good thing, at W. M. Urban's work on the question of values, and at B. Croce's arguments on value judgements. Ross concludes that the arguments in favour of thinking of value as an objective are no more successful than those in favour of treating it as a relation, and presents a lengthy validation of his conclusion, based largely on discussion of beauty and goodness. (shrink)

Tractatus Politico-Philosophicus (Political-Philosophical Treatise) of W. Julian Korab-Karpowicz proposes a new idea-system. Ideas concerning different topics related to politics are introduced. The work aims to establish the principles of good governance and of a happy society, and to open up new directions for the future development of humankind. It is also in part a critique of the epistemology of early Wittgenstein as presented in his Tractatus Logico-Philosophicus. It argues that one can speak about politics and ethics with sense, and that (...) political philosophy is still a viable enterprise for us. This explanation is provided in response to the review of Tractatus Politico-Philosophicus by Katarzyna Heremska and a critical note of Pawel Kloczowski that were both published in Argument 6(1), 2016. (shrink)

By an incomplete sentence we shall understand a declarative sentence that can be used, without variation in its meaning, to make different statements in different contexts. Although the point deserves supporting argument, which we will not provide, sentences whose grammatical subjects are indexical expressions or demonstratives are obvious, plausible examples of incomplete sentences. Uttered in one context the sentence ‘He is ill’ may be used to make one statement, for example, that George is ill, while in another context the very (...) same sentence may be used to make a quite different statement, for example, that Paul is ill. (shrink)

As late as the mid-1970s the term ‘ false consciousness ’ was still applied by critical social theorists to instances of ideological delusion. Yet, in the wake of the postmodernist revolution and its neo-Nietzschean declaration of all truths to be merely truth-effects, a concept of false consciousness appeared impossible to sustain. Drawing on an incident in the history of South African politics, this article reconsiders the ways in which a concept of false consciousness, built upon a representational model of truth (...) and falsity, might assist us in explaining the dynamics of ideological contestation. (shrink)

W. B. Gallie; I *—The Presidential Address: Rationality and the Use of Force, Proceedings of the Aristotelian Society, Volume 71, Issue 1, 1 June 1971, Pages 1–.

In An Ethic of Trust: Mutual Autonomy and the Common Will to Live, W. Royce Clark uses the work of theologians and philosophers Albert Schweitzer, Friedrich Nietzsche, and John Rawls to create an inclusive ethic in which both the religious and non-religious will have equal freedom and stability.

When the "human sciences" in the West followed the physical sciences in the 18th and 19th centuries with new measurements, methods, and language, the "metaphysics of infinity" lost its credibility. The response of Western religions was to retrench in a stronger authoritarianism, especially by the last half of the 19th century. While the new human sciences were being extended even to study the history and philosophy of religions, those religions themselves placed more emphasis on their understanding of the Absolute or (...) Unquestionable. That split became a burden both to those who were religious and to those who were not. Here, W. Royce Clark argues that humanity's survival may depend on the development of a universal or inclusive ethic in which religions move beyond their Absolutes. Clark poses the sharp challenge: Are religious communities ready to abandon the foundations that until now they have insisted were the only adequate foundation for ethics? (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Women Poets and the Origin of the Greek Hexameter W. ROBERT CONNOR A very considerable question has arisen, as to what was the origin of poetry. —Pliny the Elder, Natural History 7.57 i. a road trip with pausanias Tennyson called the dactylic hexameter “the stateliest measure / ever moulded by the lips of man,” but he did not say whose lips first did the moulding. Despite much arguing we (...) do not know for sure where it came from, or who developed it. But some ancient Greeks did, or thought they did. For them, the hexameter came with a personal history, a legendary one to be sure, but no less powerful for that. The legend, moreover, started with a woman poet. The trail that leads to this is a long and devious one, but a good guide presents himself, Pausanias, author of A Greek Guidebook (Hellados Periegesis). Although he is now valued primarily as a quarry cordoned off for topographers and archaeologists, he had a strong interest in poets, especially women ones. These include Sappho, of course, the valiant Telesilla of Argos of whom he tells two stories (2.20.8, and 2.35.2), and Praxilla of Sicyon (3.13.5). In his account of Boeotia he mentions another woman poet of the fifth century before our era, Corinna of Tanagra. She, he says (9.22.3f.), once defeated Pindar in a poetry contest, and was, he thinks, the most beautiful woman of her time, judging by a statue of her he had seen. He also speaks of Myro of Byzantium (9.5.8). Pausanias was confident that she had written epic and elegiac verse. arion 27.2 fall 2019 Pausanias chooses, moreover, to end his work with an enigmatic story about another woman poet, Anyte of Tegea. He had not mentioned her in his lengthy account of her home town in the mountains of Arcadia, but now her story seems to speak to him. It is set at a ruined shrine of Asclepius near the port town of Naupactus. The reason that Anyte had come there was a strange one: She had had, he says, a vision—not a dream, but a real vision—at the end of which she found herself holding a sealed tablet addressed to one Phalysius of Naupactus. He was, it turns out, a prosperous citizen of the town, but was going blind. Anyte made the journey, surely not an easy one, and delivered the tablet. As soon as Phalysius broke its seal, he found he could make out the letters—a wondrous improvement in his vision, which he attributed to the healing god Asclepius. That is why he built the shrine that Pausanias saw many years later. And the content of the message? Give Anyte two thousand gold staters. Phalysius did so. End of story. End of the Guidebook. So, indeed, Pausanias paid attention to women poets, and there were plenty of them to pay attention to. There is good evidence of a vibrant tradition of women’s poetry among the ancient Greeks. Pausanias is not the only writer of his time to note this. In fact, if on his journeys Pausanias had encountered the well-travelled rhetorician Tatian, the two near-contemporaries could have challenged one another to see who could name the largest number of women poets and identify the sculptors of statues honoring them. Tatian might have won, to judge from his Address to the Greeks, ch 33. But then, Tatian had an agenda. As a convert to a “barbarian” cult he wanted to show that his co-religionists had precedents for respecting women’s intellects: “My object in referring to these women is, that you may not regard as something strange what you find among us, and that, comparing the statues which are before your eyes, you may not treat the women with scorn who among us pursue philosophy.” So much for the Christian Tatian. Did Pausanias have an agenda, too? His stories of two women from Delphi, Boio 86 women poets and the origin of the greek hexameter and Phemonoe, help answer that question. ii. boio’s disruptive poetry when pausanias’s work describes Delphi, the plot thickens. In... (shrink)