The Fraenkel-Mostowski method has been widely used to prove independence results among weak versions of the axiom of choice. In this paper it is shown that certain statements cannot be proved by this method. More specifically it is shown that in all Fraenkel-Mostowski models the following hold: 1. The axiom of choice for sets of finite sets implies the axiom of choice for sets of well-orderable sets. 2. The Boolean prime ideal theorem implies a weakened form of Sikorski's theorem.

Let NBG be von Neumann-Bernays-Gödel set theory without the axiom of choice and let NBGA be the modification which allows atoms. In this paper we consider some of the well-known class or global forms of the wellordering theorem, the axiom of choice, and maximal principles which are known to be equivalent in NBG and show they are not equivalent in NBGA.

We show that it is not possible to construct a Fraenkel-Mostowski model in which the axiom of choice for well-ordered families of sets and the axiom of choice for sets are both true, but the axiom of choice is false.

In 1945–46, representatives of the U.S. government made similar discoveries in both Germany and Japan, unearthing evidence of unethical experiments on human beings that could be viewed as war crimes. The outcomes in the two defeated nations, however, were strikingly different. In Germany, the United States, influenced by the Canadian physician John Thompson, played a key role in bringing Nazi physicians to trial and publicizing their misdeeds. In Japan, the United States played an equally key role in concealing information about (...) the biological warfare experiments and in securing immunity from prosecution for the perpetrators. The greater force of appeals to national security and wartime exigency help to explain these different outcomes. (shrink)

This book is a collection of secondary essays on America's most important philosophic thinkers—statesmen, judges, writers, educators, and activists—from the colonial period to the present. Each essay is a comprehensive introduction to the thought of a noted American on the fundamental meaning of the American regime.

We study the relationship between the countable axiom of choice and the Tychonoff product theorem for countable families of topological spaces.

We find properties of topological spaces which are not shared by disjoint unions in the absence of some form of the Axiom of Choice.

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

We investigate the set theoretical strength of some properties of normality, including Urysohn's Lemma, Tietze-Urysohn Extension Theorem, normality of disjoint unions of normal spaces, and normality of Fσ subsets of normal spaces.

The axiom of multiple choice implies that metric spaces are paracompact but the reverse implication cannot be proved in set theory without the axiom of choice.

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

We study the relationship between various properties of the real numbers and weak choice principles.

We study the Tychonoff Compactness Theorem for several different definitions of a compact space.

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

We provide a general criterion for Fraenkel–Mostowski models of $${\textsf{ZFA}}$$ (i.e. Zermelo–Fraenkel set theory weakened to permit the existence of atoms) which implies “every linearly ordered set can be well ordered” ( $${\textsf{LW}}$$ ), and look at six models for $${\textsf{ZFA}}$$ which satisfy this criterion (and thus $${\textsf{LW}}$$ is true in these models) and “every Dedekind finite set is finite” ( $${\textsf{DF}}={\textsf{F}}$$ ) is true, and also consider various forms of choice for well-ordered families of well orderable sets in these (...) models. In Model 1, the axiom of multiple choice for countably infinite families of countably infinite sets ( $${\textsf{MC}}_{\aleph _{0}}^{\aleph _{0}}$$ ) is false. It was the open question of whether or not such a model exists (from Howard and Tachtsis “On metrizability and compactness of certain products without the Axiom of Choice”) that provided the motivation for this paper. In Model 2, which is constructed by first choosing an uncountable regular cardinal in the ground model, a strong form of Dependent choice is true, while the axiom of choice for well-ordered families of finite sets ( $${\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{fin}}}$$ ) is false. Also in this model the axiom of multiple choice for well-ordered families of well orderable sets fails. Model 3 is similar to Model 2 except for the status of $${\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{fin}}}$$ which is unknown. Models 4 and 5 are variations of Model 3. In Model 4 $${\textsf{AC}}_{\textrm{fin}}^{{\textsf{WO}}}$$ is true. The construction of Model 5 begins by choosing a regular successor cardinal in the ground model. Model 6 is the only one in which $$2{\mathfrak {m}} = {\mathfrak {m}}$$ for every infinite cardinal number $${\mathfrak {m}}$$. We show that the union of a well-ordered family of well orderable sets is well orderable in Model 6 and that the axiom of multiple countable choice is false. (shrink)

We study statements about countable and well-ordered unions and their relation to each other and to countable and well-ordered forms of the axiom of choice. Using WO as an abbreviation for “well-orderable”, here are two typical results: The assertion that every WO family of countable sets has a WO union does not imply that every countable family of WO sets has a WO union; the axiom of choice for WO families of WO sets does not imply that the countable union (...) of countable sets is WO. (shrink)

We study conditions for a topological space to be metrizable, properties of metrizable spaces, and the role the axiom of choice plays in these matters.

This is a continuation of [2]. We study the Tychonoff Compactness Theorem for various definitions of compactness and for various types of spaces . We also study well ordered Tychonoff products and the effect that the multiple choice axiom has on such products.

We study the relationships between definitions of compactness in topological spaces and the roll the axiom of choice plays in these relationships.

Paul C. Taylor provides an accessible guide to a well-travelled but still-mysterious area of the contemporary social landscape. The result is the first philosophical introduction to the field of race theory and to a non-biological and situational notion of race. Provides the first philosophical introduction to the field of race theory. Outlines the main features and implications of race-thinking; asks questions such as: What is race-thinking? Don’t we know better than to talk about race now? Are there any races? (...) What is it like to have a racial identity? Engages with the ideas of such important figures as Linda Alcoff, K. Anthony Appiah, W.E.B. Du Bois, Howard Winant, and Naomi Zack. Explores the enduring significance of race in relation to culture, personal relationships and social justice. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:American Journal of Philology 124.2 (2003) 306-309 [Access article in PDF] Philippe Rouet. Approaches to the Study of Attic Vases: Beazley and Pottier. Trans. Liz Nash. Oxford Monographs on Classical Archaeology. Oxford: Oxford University Press, 2001. xiii + 167 pp. 21 black and white plates. Cloth, $74. This monograph examines the development of two major approaches in the study of Greek vase painting by focusing on a comparison of (...) the work of Sir John D. Beazley with that of Edmond Pottier. Beazley, an English scholar, spent his scholarly life attributing vases to various formerly unnamed artists based on [End Page 306] their style of drawing, thereby ascertaining for the first time the artistic history and stylistic development of Attic black-figure and red-figure vase painting. Pottier, a French scholar, founded the Corpus Vasorum Antiquorum, an international series whose original goal was to publish all ancient clay vases. Each scholar's interests contrasted with the other's in many ways. Beazley's work focused more on the object and Athens, while Pottier was opposed to the study of individual artists and was particularly interested in the relationship of the origins of Greek civilization to the East and the broader context of the vessels.The book's introduction sets the historical and intellectual background of Beazley and Pottier. Areas covered include the early history of collecting, initial attempts at defining the works of individual Greek vase painters, some influential early catalogues of Greek vases (e.g., those of Hamilton's collection), and the role that the copying of motifs from ancient pottery had in spreading the interest in Greek vase painting in the late eighteenth and early nineteenth centuries, particularly the part played by the Wedgwood workshop. I miss here any reference to the fine work of Brian A. Sparkes on these subjects (The Red and the Black: Studies in Greek Pottery[London and New York, 1996] 34-63).The first chapter considers the most influential, immediate predecessors of Beazley and shows how their work formed the intellectual framework for his accomplishment. They are Wilhelm Klein, whose important study of and signatures on Attic vases still serves as the basis for our interpretation of them; Paul Hartwig, who was the first to attribute vases to an anonymous master; and Adolph Furtwängler, who in collaboration with Karl Reichhold initiated the monumental three-volume work, Griechische Vasenmalerei, which featured exceptionally accurate drawings of vase paintings. The author astutely notes that Furtwängler's work on attributing Roman copies of Greek sculpture to artists was likewise influential in augmenting the interest in style and artistic personalities. The intertwined relationship between connoisseurship, collectors, and the art market is touched on briefly in the discussions of Hartwig and Edward Warren. It might also have been noted here that the wording and style of Beazley's earliest work on the Kleophrades Painter clearly owes its inspiration to Hartwig.Pottier's early life and career are the focus of the next chapter. This was the time when museum catalogues and encyclopedic dictionaries were for many the signs of significant national scholarship. Indeed, after France's loss in 1870 in the Franco-Prussian War, Pottier and many of his contemporary compatriots sought to affirm the intellectual vitality of their nation by initiating and completing such projects. Pottier's contribution was to publish Vases antiques du Louvre, in which photography was used for the first time on a large scale to illustrate the vases, and to serve as one of the editors of the Dictionnaire des antiquités grecques et romaines, in which he was also responsible for many of the entries on vases.Chapter 3 returns to the subject of connoisseurship and the two non-classical archaeologists whose methodology bear clear resemblances to Beazley's. The first of them, Giovanni Morelli, was the first to attempt a scientific methodology for attributing paintings to unknown artists by examining the minor details [End Page 307] of drawings in order to discover the artistic language of the painters. Rouet concludes that Morelli paved the way for Beazley's work, for... (shrink)

Les débuts pathologiques de l'aventure américaine -- Prélude à l'émigration norvégienne vers l'Amérique : Kierkegaard en Norvège au XIXe siècle -- Les travaux remarquables de David Swenson -- Les choix théoriques de Walter Lowrie, ou comment Søren Kierkegaard fut métamorphosé en "S.K." -- Quelques interprétations anglophones marquantes : Collins, Thomte, Thomas, Holmer, Sponheim (James Collins, Reidar Thomte, John Heywood Thomas, Paul Holmer, Paul Sponheim--interprète de Kierkegaard) -- Le Kierkegaard de Howard Vincent Hong et Edna Hong : un (...) Kierkegaard-phare pour les recherches à venir. (shrink)

Although physicalism has been the dominant position in recent work in the philosophy of mind, this dominance has not prevented a small but growing number of philosophers from arguing that physicalism is untenable for several reasons: both ontologically and epistemologically it cannot reduce mentality to the realm of the physical, and its attempts to reduce subjectivity to objectivity have thoroughly failed. The contributors to After Physicalism provide powerful alternatives to the physicalist account of the human mind from a dualistic point (...) of view and argue that the reductive and naturalistic paradigm in philosophy has lost its force. The essays in this collection all firmly engage in a priori metaphysics. Those by Uwe Meixner, E. J. Lowe, John Foster, Alvin Plantinga, and Richard Swinburne are concerned with ways to establish the truth of dualism. Essays by William Hasker, A. D. Smith, and Howard Robinson deal with the relation between physicalism and dualism. Benedikt Paul Göcke argues that the “I” is not a particular and Stephen Priest that “I have to understand myself not as a thing but as no-thing-ness.” In the final essay, Thomas Schärtl argues that there are limits to dualism as indicated by the concept of resurrection. By including two classical essays by Plantinga and Swinburne, the volume conveniently brings together some of the best and the newest thinking in making the philosophical case for dualism. "Seven of these essays are by eminent philosophers: Lowe, Foster, Plantinga, Swinburne, Hasker, Smith, and Robinson, each recapitulating his well-known position in the debate. To have these seven essayists together under one cover constitutes a remarkable book, which can be used as a textbook in philosophy of mind as well as in philosophy of religion courses, and which also opens up the debate in an original way among colleagues at an advanced level." —Fergus Kerr, University of Edinburgh. (shrink)

Paul E. Griffiths argues that most research on the emotions has been as misguided as Aristotelian efforts to study "superlunary objects" - objects...

Many of the most important properties of human groups – including properties that may give one group an evolutionary advantage over another – are properly defined only at the level of group organization. Yet at present, most work on the evolution of culture has focused solely on the transmission of individual-level traits. I propose a conceptual extension of the theory of cultural evolution, particularly related to the evolutionary competition between cultural groups. The key concept in this extension is the emergent (...) group-level trait. This type of trait is characterized by the structured organization of differentiated individuals and constitutes a unit of selection that is qualitatively different from selection on groups as defined by traditional multilevel selection theory. As a corollary, I argue that the traditional focus on cooperation as the defining feature of human societies has missed an essential feature of cooperative groups. Traditional models of cooperation assume that interacting with one cooperator is equivalent to interacting with any other. However, human groups involve differential roles, meaning that receiving aid from one individual is often preferred to receiving aid from another. In this target article, I discuss the emergence and evolution of group-level traits and the implications for the theory of cultural evolution, including ramifications for the evolution of human cooperation, technology, and cultural institutions, and for the equivalency of multilevel selection and inclusive fitness approaches. (shrink)

Though Frege was interested primarily in reducing mathematics to logic, he succeeded in reducing an important part of logic to mathematics by defining relations in terms of functions. By contrast, Whitehead & Russell reduced an important part of mathematics to logic by defining functions in terms of relations (using the definite description operator). We argue that there is a reason to prefer Whitehead & Russell's reduction of functions to relations over Frege's reduction of relations to functions. There is an interesting (...) system having a logic that can be properly characterized in relational but not in functional type theory. This shows that relational type theory is more general than functional type theory. The simplification offered by Church in his functional type theory is an over-simplification: one can't assimilate predication to functional application.<br>. (shrink)