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)

The Dual State, first published in 1941, remains one of the most erudite books on the legal origins of democracy and dictatorship. It provided the first comprehensive analysis of the rise and nature of National Socialism, and was the only such analysis written from within Hitler's Germany. Fraenkel's concept of the dual state, being the normative state and the the prerogative state. It retains its vital relevance for the theory of democracy in the twenty-first century. The Dual State considerably (...) influenced scholars studying and working on questions of political justice in the period following World War II, particularly in the context of political and legal theory; in the domain of legal history; in the area of constitutional theory; in the context of comparative politics; and in what has become known as the field of comparative constitutional law. This republication of Fraenkel's classic work makes it once again widely available to scholars and students in the field. It includes both Fraenkel's 1974 introduction to the German second edition, never before published in English, and a new introduction by Dr Jens Meierhenrich, examining the world in which The Dual State was originally published, and the lasting legacy of this classic work. (shrink)

The lack of access to gender-affirming surgery represents a significant unmet health care need within the transgender community, frequently resulting in depression and self-destructive behavior. While some transgender people may have access to gender reassignment surgery, an overwhelming majority cannot afford facial feminization surgery. The former may be covered as a “medical necessity,” but FFS is considered “cosmetic” and excluded from insurance coverage. This demarcation between “necessity” and “cosmetic” in transgender health care based on specific body parts is in direct (...) opposition to the scientific community’s understanding of gender dysphoria and professional guidelines for transgender health. GRS affects one’s ability to function in an intimate relationship, while FFS has the same impact on social interactions an, therefore may have a far greater implication for one’s quality of life. FFS is a cost-effective intervention that needs to be covered by insurance policies. The benefits of such coverage far exceed the insignificant costs. (shrink)

There is a need to consider the impact of the new resident-hours regulations on the variety of aspects of medical education and patient care. Most existing literature about this subject has focused on the role of fatigue in resident performance, education, and health care delivery. However, there are other possible consequences of these new regulations, including a negative impact on decision ownership. Our main assumption of is that increased shift work in medicine can decrease ownership of treatment decisions and impact (...) negatively on quality of care. We review some potential components of decision ownership in treatment context and suggest possible ways in which the absence of decision ownership may decrease the quality of medical decision making. The article opens with the definition of decision ownership and the overview of some contextual factors that may contribute to the development of ownership in medical residency. The following section discusses decision ownership in medical care from the perspe... (shrink)

Many pagan, Jewish, Christian and Muslim philosophers from Antiquity to the Enlightenment made no meaningful distinction between philosophy and religion. Instead they advocated a philosophical religion, arguing that God is Reason and that the historical forms of a religious tradition serve as philosophy's handmaid to promote the life of reason among non-philosophers. Carlos Fraenkel provides the first account of this concept and traces its history back to Plato. He shows how Jews and Christians appropriated it in Antiquity, follows it (...) through the Middle Ages in both Islamic and Jewish forms and argues that it underlies Spinoza's interpretation of Christianity. The main challenge to a philosophical religion comes from the modern view that all human beings are equally able to order their lives rationally and hence need no guidance from religion. Fraenkel's wide-ranging book will appeal to anyone interested in how philosophy has interacted with Jewish, Christian, and Muslim religious traditions. (shrink)

According to both contemporary intuitions and scholarly opinion, autonomy is something specifically modern. It is certainly taken to be incompatible with religions like Islam and Judaism, if these are invested with political power. Both religions are seen as centered on a divine Law (sharî'a, viz., torah) which prescribes what we may and may not do, promising reward for obedience and threatening punishment for disobedience. Not we, but God makes the rules. This picture is in important ways misleading. There is, I (...) argue, a substantive intellectual tradition, going back to Plato's Laws, which takes the purpose of a theocracy—a community governed by God through the intermediary of a divine Law—to be promoting rational autonomy, conceived as (1) the ability to rationally determine what is in one's best interest and (2) having the motivation to live accordingly.Among the most important representatives of this intellectual tradition are medieval Muslim and Jewish philosophers. (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)

À plusieurs égards, il est vrai d’avancer que Maïmonide et Averroès poursuivent le même projet philosophique et religieux. D’autant plus que tous deux ont été décrits comme des disciples d’al-Fârâbî, le fondateur de l’école de l’aristotélisme arabe . Cependant, à première vue, leur oeuvre ne pouvait pas être moins ressemblante: Averroès n’a écrit presque exclusivement que des commentaires sur Aristote, cependant que Maïmonide n’est l’auteur d’aucune oeuvre qui appartienne à un genre philosophique dans le sens strict. Il est, d’un autre (...) côté, un commentateur distingué lui aussi - sauf qu’il n’explique pas Aristote, mais commente et met de l’ordre dans la Loi de Moïse. La question principale dont je traite dans cet article, est celle de savoir si ce rapport tellement différent à la philosophie et à l’exégèse chez Averroès et Maïmonide peut être compris comme deux façons d’implémenter le cadre conceptuel établi par al-Fârâbî. Je commence par examiner le projet d’al-Fârâbî, que j’essaie d’expliquer par deux objectifs: reprendre et continuer le projet de la philosophie antique et définir sa place dans une société, dans laquelle l’autorité de la Loi divine n’est pas mise en question. Ensuite, je montre que, tandis que l’oeuvre d’Averroès peut être comprise, dans l’ensemble, comme une continuation du projet d’al-Fârâbî, cela n’est pas vrai de la même façon pour l’oeuvre de Maïmonide qui en partie transforme al-Fârâbî de manière créative et en partie utilise des prémisses dont on ne trouvera pas la source en al-Fârâbî. La position de Maïmonide en ce qui concerne la philosophie et l’exégèse se distingue significativement de la position habituelle adoptée par les falâsifa, ce qui a eu des conséquences importantes pour la philosophie juive médiévale après Maïmonide. (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)

This volume draws a balanced picture of the Rationalists by bringing their intellectual contexts, sources and full range of interests into sharper focus, without neglecting their core commitment to the epistemological doctrine that earned ...

Arcturus is a very suitable person to tell us the ποκεíμενα of the play. During the night preceding the action he has produced the storm which is to bring about the dénouement of the plot : illorum navis longe in altum abscesserat. ego quoniam video virginem asportarier, tetuli ei auxilium et lenoni exitium simul: increpui hibernum et fluctus movi maritimos. nam Arcturus signum sum omnium acerrimum: vehemens sum exoriens, cum occido vehementior.