Russell’s paradox is purely logical in the following sense: a contradiction can be formally deduced from the proposition that there is a set of all non-self-membered sets, in pure first-order logic—the first-order logical form of this proposition is inconsistent. This explains why Russell’s paradox is portable—why versions of the paradox arise in contexts unrelated to set theory, from propositions with the same logical form as the claim that there is a set of all non-self-membered sets. Burali-Forti’s paradox, like (...) Russell’s paradox, is portable. I offer the following explanation for this fact: Burali-Forti’s paradox, like Russell’s, is purely logical. Concretely, I show that if we enrich the language \ of first-order logic with a well-foundedness quantifier W and adopt certain minimal inference rules for this quantifier, then a contradiction can be formally deduced from the proposition that there is a greatest ordinal. Moreover, a proposition with the same logical form as the claim that there is a greatest ordinal can be found at the heart of several other paradoxes that resemble Burali-Forti’s. The reductio of Burali-Forti can be repeated verbatim to establish the inconsistency of these other propositions. Hence, the portability of the Burali-Forti’s paradox is explained in the same way as the portability of Russell’s: both paradoxes involve an inconsistent logical form—Russell’s involves an inconsistent form expressible in \ and Burali-Forti’s involves an inconsistent form expressible in \. (shrink)

The year 1897 saw the publication of the first of the modern logical paradoxes. It was published by Cesare Burali-Forti, the Italian mathematician whose name it has come to bear. Burali-Forti's own formulation of the paradox was not altogether satisfactory, as he had confused well-ordered sets as defined by Cantor with what he himself called “perfectly ordered sets”. However, he soon realized his mistake, and published a note admitting the error and making the correction. He concluded (...) the note with the observation that his result could be established on the basis of the correct definition of well-ordered set as easily as for the “perfectly ordered sets” for which it had first been obtained. We shall reproduce his results in their corrected form. (shrink)

In studying the early history of mathematical logic and set theory one typically reads that Georg Cantor discovered the so-called Burali-Forti (BF) paradox sometime in 1895, and that he offered his solution to it in his famous 1899 letter to Dedekind. This account, however, leaves it something of a mystery why Cantor never discussed the paradox in his writings. Far from regarding the foundations of set theory to be shaken, he showed no apparent concern over the paradox and (...) its implications whatever. Against this account, I will argue here that in fact Cantor never saw any paradox at all, but that his conception of set at that time, and already as far back as 1883, was one in which the paradoxes cannot arise. (shrink)

Philip Jourdain put this question to Frege in a letter of 28 January 1909. Frege had, indeed, next to nothing to say about ordinals, and in this respect Bob Hale has followed the master. As I hope this chapter will show, though, the topic is worth addressing. The natural abstraction principle for ordinals combines with full, impredicative second-order logic to engender a contradiction, the so-called Burali-Forti Paradox. I shall contend that the best solution involves a retreat to a (...) predicative logic. Such a retreat has implications for other neo-Fregean theories, including the cardinal arithmetic on which Hale has focused. The discussion will touch on a topic which has been at the centre of Hale’s more recent work—namely, the interpretation of plural and higher-order quantifiers. (shrink)

After briefly reviewing the standard set-theoretic resolutions of the Burali-Forti paradox, we examine how the paradox arises in set theory formalized with plural quantifiers. A significant choice emerges between the desirable unrestricted availability of ordinals to represent well-orderings and the sensibility of attempting to refer to ‘absolutely all ordinals’ or ‘absolutely all well-orderings’. This choice is obscured by standard set theories, which rely on type distinctions which are obliterated in the setting with plurals. Zermelo's attempt ( 1930 ) (...) to secure ordinal representability of arbitrary well-orderings through relativization of quantification to set-theoretic models is reviewed and found wanting. The natural modal-structural recasting provides, it is claimed, a good repair. (shrink)

The paradox that appears under Burali-Forti’s name in many textbooks of set theory is a clever piece of reasoning leading to an unproblematic theorem. The theorem asserts that the ordinals do not form a set. For such a set would be—absurdly—an ordinal greater than any ordinal in the set of all ordinals. In this article, we argue that the paradox of Burali-Forti is first and foremost a problem about concept formation by abstraction, not about sets. We (...) contend, furthermore, that some hundred years after its discovery the paradox is still without any fully satisfactory resolution. A survey of the current literature reveals one key assumption of the paradox that has gone unquestioned, namely the assumption that ordinals are objects. Taking the lead from Russell’s no class theory, we interpret talk of ordinals as an efficient way of conveying higher-order logical truths. The resulting theory of ordinals is formally adequate to standard intuitions about ordinals, expresses a conception of ordinal number capable of resolving Burali-Forti’s paradox, and offers a novel contribution to the longstanding program of reducing mathematics to higher-order logic. (shrink)

L'historiographie des paradoxes ensemblistes attribue classiquement la "découverte" du paradoxe du plus grand ordinal au mathématicien italien Burali-Forti. Un examen attentif de ses démonstration et revirement révèle qu'il n'en est rien. Nous tenterons de dégager l'impact de la publication des Principles de Russell sur la constitution de cette version historiographique officielle, avant d'aborder la question de la définition des paradoxes et ce qui peut en motiver une conception restrictive.

El presente escrito presenta las ventajas y desventajas de la formalización matemática como una herramienta para el análisis de argumentos complejos o difusos en la filosofía. De tal forma, aquí se encuentra un recorrido histórico de algunas consideraciones del papel de las matemáticas en la búsqueda del conocimiento. Posterior a ello, se muestra cómo por medio de la teoría de conjuntos y laabstracción matemática, es posible proponer una reinterpretación de algunos textos filosóficos. Para lograr este objetivo, se presenta, a manera (...) de ejemplo, la formalización de los conceptos de “Dios”, “error” y “Creación” dados por Descartes en su libro las Meditaciones de la filosofía primera, este ejercicio permiteproponer una paradoja en donde se evidencia una posible falencia en la delimitación del concepto de Dios. Con todo ello, se demuestra por qué los enfoques matemáticos deben ser promovidos en los estudios filosóficos. (shrink)

Das angebliche Bilderverbot, das im Zweiten Gebot des Dekalogs enthalten sein soll, ist eigentlich ein Idolatrieverbot. Das heißt, dass das jüdische Gesetz nicht Bilder an sich, sondern Idole verbietet. Gewiss: Manche Bilder werden als Idole verehrt. Es gibt aber auch Bilder, die keine idolatrische Bedeutung haben, und umgekehrt Idole, die keinen bildlichen Charakter aufweisen. Die Untersuchung dieser komplexen Zusammenhänge ist das Hauptziel des vorliegenden Sammelbandes. Von unterschiedlichen theoretischen Standpunkten ausgehend, eröffnen die hier versammelten Aufsätze neue Perspektiven auf das Verhältnis zwischen (...) Bild und Idol. Mit Beiträgen von Beniamino Fortis, Asher D. Biemann, Ellen Rinner, Mario C. Schmidt, Lars Tittmar, Johannes Bennke und Agata Bielik-Robson. (shrink)

In the last decade, we have witnessed the return of one of the most controversial terms in the political lexicon: totalitarianism. What are we talking about when we define a totalitarian political and social situation? When did we start using the word as both adjective and noun? And, what totalitarian ghosts haunt the present? Philosopher Simona Forti seeks to answer these questions by reconstructing not only the genealogy of the concept, but also by clarifying its motives, misunderstandings, and the (...) controversies that have animated its current resurgence. Taking into account political theories and historical discussions, Totalitarianism especially focuses on philosophical reflections, from the question of totalitarian biopolitics to the alleged totalitarian drifts of neoliberalism. The work invites the relentless formulation of a radical question about democratic age: the possibilities it has opened up, the voids it leaves behind, the mechanisms it activates, and the "voluntary servitude" it produces. Forti argues that totalitarianism cannot be considered an external threat to democracy, but rather, one of the possible answers to those questions posed by modernity which democracies have not been able to solve. Her investigation of the uses and abuses of totalitarianism as one of the fundamental categories of 20th and 21st centuries promises to provoke much-needed discussion and debate among those in philosophy, politics, ethics, and beyond. (shrink)

As long as we care about suffering in the world, says political philosopher Simona Forti, we are compelled to inquire into the question of evil. But is the concept of evil still useful in a postmodern landscape where absolute values have been leveled and relativized by a historicist perspective? Given our current unwillingness to judge others, what signposts remain to guide our ethical behavior? Surveying the nineteenth- and twentieth-century Western philosophical debates on evil, Forti concludes that it is (...) time to leave behind what she calls "the Dostoevsky paradigm": the dualistic vision of an omnipotent monster pitted against absolute, helpless victims. No longer capable of grasping the normalization of evil in today's world—whose structures of power have been transformed—this paradigm has exhausted its explanatory force. In its place, Forti offers a different genealogy of the relationship between evil and power, one that finally calls into question power's recurrent link to transgression. At the center of contemporary evil she posits the passive attitude towards rule-following, the need for normalcy, and the desire for obedience nurtured by our contemporary mass democracies. In our times, she contends, evil must be explored in tandem with our stubborn desire to stay alive at all costs as much as with our deep need for recognition: the new modern absolutes. A courageous book, New Demons extends an original, inspiring call to ethical living in a biopolitical age. (shrink)

It is well known that the validity of Choice Principles is problematic in non-standard Set Theories which do not abide by the Limitation of Size Principle. In this paper we discuss the consistency of various Choice Principles with respect to the Generalized Positive Comprehension Principle . The Principle GPC allows to take as sets those classes which can be specified by Generalized Positive Formulae, e.g. the universe. In particular we give a complete characterization of which choice principles hold in Hyperuniverses. (...) Hyperuniverses are structures which arose independently in Non-well-founded Set Theory and in Mathematical Semantics of Concurrent Programming Languages and are hitherto the only existing models of GPC. Hyperuniverses are naturally endowed with a κ-compact uniform κ-topology and are uniformly isomorphic to their exponential space, i.e. the space of their closed subsets endowed with the Exponential Uniformity. (shrink)

This essay focuses on the relationship between biopolitics and race theory. Starting from Foucault, many authors have considered totalitarian anti-Semitism as a depravity of biologism. This essay would like to challenge this all-too-simple positivist, materialist, and evolutionist picture of biopolitics in the Third Reich. It examines another "tradition" of racial theories, central to National Socialism, much closer to the revered Western philosophical tradition than Darwinism ever was. This kind of racism presents itself as the authentic heir of that "Metaphysics of (...) Form," which traces its roots back to classical antiquity, in particular to Plato's work. Through the analysis of some Platonist and racist texts the essay tries to point out the ambivalences that connect some of the assumptions of our philosophical tradition to Nazi totalitarianism. (shrink)

A possible world is a junky world if and only if each thing in it is a proper part. The possibility of junky worlds contradicts the principle of general fusion. Bohn (2009) argues for the possibility of junky worlds, Watson (2010) suggests that Bohn‘s arguments are flawed. This paper shows that the arguments of both authors leave much to be desired. First, relying on the classical results of Cantor, Zermelo, Fraenkel, and von Neumann, this paper proves the possibility of junky (...) worlds for certain weak set theories. Second, the paradox of Burali-Forti shows that according to the Zermelo-Fraenkel set theory ZF, junky worlds are possible. Finally, it is shown that set theories are not the only sources for designing plausible models of junky worlds: Topology (and possibly other "algebraic" mathematical theories) may be used to construct models of junky worlds. In sum, junkyness is a relatively widespread feature among possible worlds. (shrink)

When considered in the field of aesthetics, the Bilderverbot (biblical ban on pictures) is received with a variety of attitudes ranging between the extremes of Kant’s praise and Hegel’s criticism. Despite being at odds with each other, Kant’s and Hegel’s interpretations suffer from the same theoretical flaw: both assume that the pictures the Bible talks about are representations related to their objects by way of reference. This assumption is proven wrong in the pars destruens of this essay, in which it (...) will be argued that using the notion of ‘representation’ as a key to the reading of the Bilderverbot is a misconception that leads to an anachronistic and partial view of the matter. The concept of ‘gaze,’ elaborated by Régis Debray and other scholars of visual studies, is introduced in the pars construens of this essay to show that the biblical prohibition is actually aimed at a different kind of pictures, to wit at presentifications, which, instead of referring to their objects in a representational way, are rather thought to embody and dominate them. In conclusion, through Nietzsche’s insights into the reasons behind religious cults, the point is finally made that the notion of ‘power’ surpasses that of ‘representation’ in capturing and accounting for the authentic meaning of the Bilderverbot. (shrink)

The proof of Lemma 5 in our paper “Choice Principles in Hyperuniverses” [3], contains an error. In the present note we show that the statement of that lemma is false and hence the Axiom of Choice fails in all κ-hyperuniverses, for uncountable κ. However, a weaker version of Lemma 5 can be proved, which implies that the Linear Ordering Principle holds in all κ-metric κ-hyperuniverses.

As long as we care about suffering in the world, says political philosopher Simona Forti, we are compelled to inquire into the question of evil. But is the concept of evil still useful in a postmodern landscape where absolute values have been leveled and relativized by a historicist perspective? Given our current unwillingness to judge others, what signposts remain to guide our ethical behavior? Surveying the nineteenth- and twentieth-century Western philosophical debates on evil, Forti concludes that it is (...) time to leave behind what she calls "the Dostoevsky paradigm": the dualistic vision of an omnipotent monster pitted against absolute, helpless victims. No longer capable of grasping the normalization of evil in today's world—whose structures of power have been transformed—this paradigm has exhausted its explanatory force. In its place, Forti offers a different genealogy of the relationship between evil and power, one that finally calls into question power's recurrent link to transgression. At the center of contemporary evil she posits the passive attitude towards rule-following, the need for normalcy, and the desire for obedience nurtured by our contemporary mass democracies. In our times, she contends, evil must be explored in tandem with our stubborn desire to stay alive at all costs as much as with our deep need for recognition: the new modern absolutes. A courageous book, New Demons extends an original, inspiring call to ethical living in a biopolitical age. (shrink)