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)

Foundations of Set Theory discusses the reconstruction undergone by set theory in the hands of Brouwer, Russell, and Zermelo. Only in the axiomatic foundations, however, have there been such extensive, almost revolutionary, developments. This book tries to avoid a detailed discussion of those topics which would have required heavy technical machinery, while describing the major results obtained in their treatment if these results could be stated in relatively non-technical terms. This book comprises five chapters and begins with a discussion of (...) the antinomies that led to the reconstruction of set theory as it was known before. It then moves to the axiomatic foundations of set theory, including a discussion of the basic notions of equality and extensionality and axioms of comprehension and infinity. The next chapters discuss type-theoretical approaches, including the ideal calculus, the theory of types, and Quine's mathematical logic and new foundations; intuitionistic conceptions of mathematics and its constructive character; and metamathematical and semantical approaches, such as the Hilbert program. This book will be of interest to mathematicians, logicians, and statisticians. (shrink)

Mental state reasoning or theory-of-mind has been the subject of a rich body of imaging research. Although such investigations routinely tap a common set of regions, the precise function of each area remains a contentious matter. With the help of functional magnetic resonance imaging (fMRI), we sought to determine which areas are involved when processing mental state or intentional metarepresentations by focusing on the relational aspect of such representations. Using non-intentional relational representations such as spatial relations between persons and between (...) objects as a contrast, the results ascertained the involvement of the precuneus, the temporal poles, and the medial prefrontal cortex in the processing of intentional representations. In contrast, the anterior superior temporal sulcus and the left temporo-parietal junction were implicated when processing representations that refer to the presence of persons in relational contexts in general. The right temporo-parietal junction, however, was specifically activated for persons entering spatial relations. The level of representational complexity, a previously unexplored factor, was also found to modulate the neural response in some brain regions, such as the medial prefrontal cortex and the right temporo-parietal junction. These findings highlight the need to take into account the critical roles played by an extensive network of neural regions during mental state reasoning. (shrink)

Einleitung. Das “Auswahlaxiom” oder “multiplicative axiom” fordert in der gewöhnlichen, von B. Russell und Zermelo angegebenen Fassung, daß zu jeder Menge M, deren Elemente paarweise fremde und nicht-leere Mengen sind, mindestens eine “Auswahlmenge” existiere, die mit jedem Element von M genau éin Element gemeinsam hat. Die nächstliegende und mehrfach verwendete Methode, um ein schwächeres Postulat als die vorstehende Fassung zu formulieren, besteht darin, daß man entweder über die Mächtigkeit der Menge M, oder über die Mächtigkeiten ihrer Elemente, oder über beides (...) gleichzeitig, einschränkende Bedingungen macht, also nur in diesen eingeschränkten Fällen die Existenz einer Auswahlmenge verlangt. Hinsichtlich der Mächtigkeit von M ist da sogleich die Bemerkung zu machen, daß sie jedenfalls transfinit (genauer: nicht endlich im induktiven Sinn) anzusetzen ist. Vermöge allgemeiner logischer Sachverhalte und des (eventuell zu beweisenden) Prinzips, das zu einem gegebenen Objekt a die Bildung der Menge {a} erlaubt, ist nämlich die Existenz einer Auswahlmenge von M beweisbar, falls M die Mächtigkeit 1 besitzt, und diese Beweisbarkeit überträgt sich mittels gewöhnlicher Induktion auf den Fall einer beliebigen endlichen Mächtigkeit. Dagegen ist die Existenz einer Auswahlmenge nicht mehr beweisbar, wenn (bei transfinitem M) die Elemente von M endliche Mengen mit Mächtigkeiten > 1 sind, z. B. also schon in dem Fall, wo M eine transfinite Menge von Paaren ist—natürlich “im allgemeinen,” d. h. wenn die Elemente von M nicht eine besondere Beschaffenheit haben, die die Auszeichnung gewisser Elemente in ihnen ermöglicht. Die Tatsache dieser Nichtbeweisbarkeit, d. h. der Unabhängigkeit des Auswahlaxioms sogar in diesem einfachsten Fall, ist vor längerer Zeit auf der Basis des Zermeloschen Axiomensystems der Mengenlehre nachgewiesen worden. (shrink)

Die Frühe Neuzeit hat in den Geschichts- und Kulturwissenschaften ein eigenes Profil gewonnen. Die Buchreihe Frühe Neuzeit dient der Grundlagenforschung in Gestalt von Editionen, Monographien und Sammelbänden. Sie strebt nicht die großräumige Überschau an, die vorschnelle Synthese oder prätentiöse Konstruktion, sondern nimmt den Umweg über die Arbeit am Detail und die Erkundung verschütteter Traditionszusammenhänge. Ein besonderer Akzent liegt auf Untersuchungen, welche die Grenzen der Fachdisziplin überschreiten.

Most of the early proofs of the decidability or completeness of certain mathematical theories were based on the method of eliminations of quantifiers. Various more recent results on completeness were obtained independently of such procedures. However, it is shown in the present paper that, conversely, the completeness of a mathematical theory will in certain circumstances entail the existence of an elimination method. The proof involves the application of the extended first ε‐theorem of Hilbert‐Bernays.ZusammenfassungDie meisten früheren Beweise der Vollständigkeit oder Entscheidbarkeit (...) gewisser mathematischer Theorien benutzten die sogenannte Quantoren‐eliminierungsmethode. Dagegen sind einige neuere Ergebnisse über Voll‐ständigkeit unabhängig von diesem Verfahren. Es ist deshalb von Interesse festzustellen, dass umgekehrt die Vollständigkeit einer mathematischen Theorie unter Umständen die Existenz einer Eliminierungsmethode fürdiese Theorie nach sich zieht. Der Beweis benutzt das erweiterte erste ε‐Theorem von Hilbert‐Bernays.SommaireBeaucoup de démonstrations concernant la complétude de certaines théories mathématiques, ou concernant l'existence d'un procédé de décision pour une telle théorie, sont basées sur des méthodes d'élimination de quantificateurs. Or, il y a des résultats plus récents sur la complétude de certaines théories qui ne dépendent pas de telles méthodes d'élimination. Toutefois on démontre ici que la complétude d'une théorie mathématique entraîne sous certaines conditions l'existence d'une méthode d'élimination. Au cours de la démonstration on se sert d'un des théorèmes‐ε de Hilbert‐Bernays. (shrink)

The question of the independence of the axioms of the theory of sets has been dealt with in a number of works, although not in a final manner. The writer will be concerned solely with the axiomatic system of Zermelo and Fraenkel, and only with that feature of the system whereby all the objects of the underlying domain are sets (so that there is no difference between objects in general and sets in particular).A special place among the axioms is (...) occupied by a purely relational one, an axiom of definiteness, which establishes the character of equality within the system.Zermelo introduces equality intensionally—if two symbolsxandyrepresent the same object, we write = (x, y). According to this there is no necessity for axioms to assure the interchangeability of equal objects as arguments of the primitive relationϵ(, ). For under the intensional interpretation it is clear that:Similarly it is clear that there is no necessity for a separate axiom that will assure to equality the properties of an equivalence relation. Therefore Zermelo introduces only the followingaxiom of extensionality:(if one set is a subset of another set, and the second set is also a subset of the first, then the two sets are equal). (shrink)

The article provides an overview of Jewish Life-of-Jesus research from Abraham Geiger to Ernst Ludwig Ehrlich. Julius Wellhausen's assessment that Jesus was not Christian but Jewish encountered a Jewish community that was striving for civic equality in the course of the Enlightenment and that saw itself impaired by the idea of the,,Christian state". The ensuing Jewish concern with the central figure of the New Testament was not of fundamental nature, but rather followed from an apologetic impulse: the wish to (...) participate in general society without having to give up Jewish identity. Since then, many Jewish thinkers of the modern era have studied Jesus. The essay outlines the history of,,bringing Jesus home" to Judaism, which has been observed since the nineteenth century. Jesus returns as exemplary Jew, as hortatory prophet, as revolutionary and freedom fighter, as big brother and messianic Zionist. The foremost intention though was that Jews wanted to remain Jews and nevertheless be part of Christian society. How fortunate, therefore, that Jesus was Jewish. (shrink)

Felix Klein and Abraham Fraenkel each formulated a criterion for a theory of infinitesimals to be successful, in terms of the feasibility of implementation of the Mean Value Theorem. We explore the evolution of the idea over the past century, and the role of Abraham Robinson's framework therein.

From the perspective of Heilsgeschichte, Gerhard von Rad saw clearly that Genesis 22 deals with the possible annihilation of the covenant promise. A fresh approach to Genesis corroborates this view and demonstrates that innerbiblical exegesis has shaped the message of Genesis 22.

The objective of this article is to provide a formalism to deal with the special theory of relativity (STR, in short) as riewed by Reichenbach, according to which STR involves an ineradicableconventionality of simultaneity. One of the two postulates of STR asserts that, in empty space, the one-way speed of light relative to inertial frames is constant. Experimental evidence, however, is related to the constancy of the round-trip speed of light and has no bearing on one-way speeds. Following Reichenbach's viewpoint, (...) we relax the second postulate of STR, abandoning the constancy of the one-way speed of light to the more realistic one asserting the constancy of the round-trip speed of light. This, in turn, results in a formalism to deal with Reichenbach's special theory of relativity (RSTR, in short) in which the two one-way speeds of light in empty space, C±, in the two senses of a round-trip are arbitrarily selected in such a way that their harmonic mean is the measurable round-trip speed of light, c. Experimentally, RSTR and STR are indistinguishable and, hence, represent the same physical theory. It is only the formalism that we use to deal with STR which is extended in RSTR to accommodate the immeasurability of one-way velocities. The usefulness of the proposed formalism to study special relativity is demonstrated. (shrink)

In early April 1911 Albert Einstein arrived in Prague to become full professor of theoretical physics at the German part of Charles University. It was there, for the first time, that he concentrated primarily on the problem of gravitation. Before he left Prague in July 1912 he had submitted the paper "Relativität und Gravitation: Erwiderung auf eine Bemerkung von M. Abraham" in which he remarkably anticipated what a future theory of gravity should look like. At the occasion of the (...) Einstein-in-Prague centenary an international meeting was organized under a title inspired by Einstein's last paper from the Prague period: "Relativity and Gravitation, 100 Years after Einstein in Prague". The main topics of the conference included: classical relativity, numerical relativity, relativistic astrophysics and cosmology, quantum gravity, experimental aspects of gravitation, and conceptual and historical issues. The conference attracted over 200 scientists from 31 countries, among them a number of leading experts in the field of general relativity and its applications. This volume includes abstracts of the plenary talks and full texts of contributed talks and articles based on the posters presented at the conference. These describe primarily original results of the authors. Full texts of the plenary talks are included in the volume "General Relativity, Cosmology and Astrophysics--Perspectives 100 Years after Einstein in Prague", editions. J. Bičák and T. Ledvinka, published also by Springer Verlag. (shrink)

In the present paper the story is told of the brief and far from tranquil encounter of L.E.J. Brouwer and A. Fraenkel. The relationship which started in perfect harmony, ended in irritation and reproaches.The mutual appreciation at the outset is beyond question. All the more deplorable is the sudden outbreak of an emotional disagreement in 1927. Looking at the Brouwer–Fraenkel episode, one should keep in mind that at that time the so-called Grundlagenstreit was in full swing. An emotional (...) man like Brouwer, who easily suffered under stress, was already on edgewhen Fraenkel's book Zehn Vorlesungen Über die Grundlegung der Mengenlehre, [Fraenkel 1927] was about to appear.With the Grundlagenstreit reaching a level of personal abuse unusual in the quiet circles of pure mathematics, Brouwer was rather sensitive, where the expositions of his ideas were concerned. So when he thought that he detected instances of misconception and misrepresentation in the case of his intuitionism, he felt slighted. We will mainly look at Brouwer's reactions. since the Fraenkel letters have not been preserved.The late Mrs. Fraenkel kindly put the Brouwer letters that were in her possession at my disposal. I am grateful to the Fraenkel family for the permission to use the material.I am indebted to Andreas Blass for his valuable suggestions and corrections.Abraham Fraenkel was one of the first non-partisan mathematicians, if not the first, who developed a genuine interest in intuitionism. (shrink)

The growth of global value chains (GVCs) and the emergence of novel forms of value chain governance pose two questions for normative business ethics. First, how should we conceptualize the relationships between members of a GVC? Second, what ethical implications follow from these relationships, both with respect to interactions between GVC members and with respect to achieving broader transnational governance goals? We address these questions by examining the emergence of transnational eco-labeling as an increasingly prominent form of GVC governance that (...) is redefining the relationship between nominally independent firms. On the first question, we argue that GVCs occupy a middle ground between intrafirm and interfirm transactions, thereby posing a challenge to theoretical frameworks that attempt to apply ethical standards based on transaction types. On the second question, we argue that this unique institutional status leads to a range of novel ethical considerations and dilemmas for GVC members. Lead firms, their suppliers, and third-party standard-setters all confront new ethical quandaries when third-party eco-labeling is introduced to a GVC. The nature of these quandaries means that, while GVCs may be well-positioned to serve as instruments of transnational governance, they are frequently not well-oriented to do so. Our analysis marks an initial attempt to map the ethical considerations that apply to members of a GVC, and in doing so, places the literature on normative business ethics and transnational governance in a closer conversation. (shrink)

Gyrogroup theory [A. A. Ungar, Found. Phys. 27, 881–951 (1997)] enables the study of the algebra of Einstein's addition to be guided by analogies shared with the algebra of vector addition. The capability of gyrogroup theory to capture analogies is demonstrated in this article by exposing the relativistic composite-velocity reciprocity principle. The breakdown of commutativity in the Einstein velocity addition ⊕ of relativistically admissible velocities seemingly gives rise to a corresponding breakdown of the relativistic composite-velocity reciprocity principle, since seemingly (i) (...) on one hand, the velocity reciprocal to the composite velocity u⊕v is −(u⊕v) and (ii) on the other hand, it is (−v)⊕(−u). But (iii) −(u⊕v)≠(−v)⊕(−u). We remove the confusion in (i), (ii), and (iii) by employing the gyrocommutative gyrogroup structure of Einstein's addition and, subsequently, present the relativistic composite-velocity reciprocity principle with the Thomas rotation that it involves. (shrink)

The growth of global value chains (GVCs) and the emergence of novel forms of value chain governance pose two questions for normative business ethics. First, how should we conceptualize the relationships between members of a GVC? Second, what ethical implications follow from these relationships, both with respect to interactions between GVC members and with respect to achieving broader transnational governance goals? We address these questions by examining the emergence of transnational eco‐labeling as an increasingly prominent form of GVC governance that (...) is redefining the relationship between nominally independent firms. On the first question, we argue that GVCs occupy a middle ground between intrafirm and interfirm transactions, thereby posing a challenge to theoretical frameworks that attempt to apply ethical standards based on transaction types. On the second question, we argue that this unique institutional status leads to a range of novel ethical considerations and dilemmas for GVC members. Lead firms, their suppliers, and third‐party standard‐setters all confront new ethical quandaries when third‐party eco‐labeling is introduced to a GVC. The nature of these quandaries means that, while GVCs may be well‐positioned to serve as instruments of transnational governance, they are frequently not well‐oriented to do so. Our analysis marks an initial attempt to map the ethical considerations that apply to members of a GVC, and in doing so, places the literature on normative business ethics and transnational governance in a closer conversation. (shrink)

The relativistic velocity composition paradox of Mocanu and its resolution are presented. The paradox, which rests on the bizarre and counterintuitive non-communtativity of the relativistic velocity composition operation, when applied to noncollinear admissible velocities, led Mocanu to claim that there are “some difficulties within the framework of relativistic electrodynamics.” The paradox is resolved in this article by means of the Thomas rotation, shedding light on the role played by composite velocities in special relativity, as opposed to the role they play (...) in Galilean relativity. (shrink)

English summary: The wide-ranging work of Ernst Fraenkel lead to the foundation of postwar political science. In his role as "American in Berlin," Fraenkel helped shape the foundation of modern comparative government theory. Fraenkel's impressive, and in retrospect exemplary, biography, from being a Jewish labor lawyer in the Third Reich to an emigrant to the United States, allowed for him to be described as a commanding figure of the young field of political science of the Adenauer era. (...) Reinhard Dorn illuminates Fraenkel's integrative state and constitutional understanding in two parts. In the first section, he presents foundations and individual components of constitutional-sociological thought, whereby he focuses particular attention on sociological and policy-sociological aspects. The second section documents the performance range of Fraenkel's constitution sociology by comparing Anglo-American and continental European constitutional thinking with its respective constitutional reality, all based on his own interdisciplinary analysis. German text. German description: Die Beschaftigung mit dem breit gefacherten Werk von Ernst Fraenkel fuehrt zu den Grundlagen der Nachkriegs-Politikwissenschaft. In seiner Rolle als Amerikaner in Berlin hat Fraenkel das Fundament der modernen vergleichenden Regierungslehre mitgestaltet. Dabei hat Fraenkels beeindruckende und im Rueckblick exemplarisch wirkende Biographie vom juedischen Arbeitsrechtler und Rechtsanwalt im Dritten Reich ueber den Emigranten in die USA zur Portalfigur der jungen Politikwissenschaft der Adenauer-Ara seine Verfassungsvergleiche erst ermoglicht.In zwei Teilen beleuchtet Reinhard Dorn Fraenkels integratives Staats- und Verfassungsverstandnis: Im ersten Abschnitt stellt er Grundlagen und einzelne Bestandteile seines verfassungssoziologischen Denkens dar, wobei er insbesondere auf soziologische und politiksoziologische Aspekte eingeht. Der zweite Abschnitt dokumentiert die Leistungsbreite von Fraenkels Verfassungssoziologie, indem - ausgehend von dessen eigenen interdisziplinaren Analysen - angloamerikanisches und kontinentaleuropaisches Verfassungsdenken der jeweiligen Verfassungsrealitat gegenuebergestellt wird. (shrink)

The obscured Thomas precessionof the special theory of relativity (STR) has been soared into prominence by exposing the mathematical structure, called a gyrogroup,to which it gives rise [A. A. Ungar, Amer. J. Phys.59,824 (1991)], and the role that it plays in the study of Lorentz groups [A. A. Ungar, Amer. J. Phys.60,815 (1992); A. A. Ungar, J. Math. Phys.35,1408 (1994); A. A. Ungar, J. Math. Phys.35,1881 (1994)]. Thomas gyrationresults from the abstraction of Thomas precession.As such, its study sheds light on (...) relativistic velocity spaces and their symmetries which are concealed in Thomas precession. In order to uncover new properties of relativistic gyrogroups, we employ in this article the group theoretic concepts of divisible groupsand two-torsion free groupsto construct midpointsin gyrogroups. Systems of successive midpoints then describe straight gyrolinesand suggest a new, natural distance function that involves a Thomas gyration. These, in turn, reveal a new, interesting geometry that underlies relativistic velocity spaces. In this resulting gyrogeometrythe straight gyrolines form geodesics under a distance function which turns out to be the Poincaré hyperbolic distance function relaxed by a Thomas gyration. These geodesics do obey the parallel axiom of Euclidean geometry. Ironically, (i) attempts to understand the parallel postulate of Euclidean geometry gave rise to hyperbolic geometry in which the parallel postulate disappears;(ii) hyperbolic geometry gave rise to Einstein's STR; (iii) Einstein's STR established the bizarre and counterintuitive relativistic effect called Thomas precession, the abstraction of which is called Thomas gyration; and (iv) Thomas gyration now repairs in this article the Poincaré distance function of hyperbolic geometry to the point where the parallel postulate reappears. (shrink)