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)

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)

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)

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- (...) class='Hi'>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 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)

Graham Priest (1994) has argued that the following paradoxes all have the same structure: Russell’s Paradox, Burali-Forti’s Paradox, Mirimanoff’s Paradox, König’s Paradox, Berry’s Paradox, Richard’s Paradox, the Liar and Liar Chain Paradoxes, the Knower and Knower Chain Paradoxes, and the Heterological Paradox. Their common structure is given by Russell’s Schema: there is a property φ and function δ such that..

Reasonning in naive set theory (with unlimited comprehension), we derive a paradox (a formal contradiction) which can be seen as a variant of the Burali-Forti paradox.

We present a semi-formal foundational theory of sorts, akin to sets, named librationism because of its way of dealing with paradoxes. Its semantics is related to Herzberger’s semi inductive approach, it is negation complete and free variables (noemata) name sorts. Librationism deals with paradoxes in a novel way related to paraconsistent dialetheic approaches, but we think of it as bialethic and parasistent. Classical logical theorems are retained, and none contradicted. Novel inferential principles make recourse to theoremhood and failure of theoremhood. (...) Identity is introduced à la Leibniz-Russell, and librationism is highly non-extensional. Π 1 1 -comprehension with ordinary Bar-Induction is accounted for (to be lifted). Power sorts are generally paradoxical, and Cantor’s Theorem is blocked as a camouflaged premise is naturally discarded. (shrink)

We are familiar with various set-theoretical paradoxes such as Cantor's paradox, Burali-Forti's paradox, Russell's paradox, Russell-Myhill paradox and Kaplan's paradox. In fact, there is another new possible set-theoretical paradox hiding itself in Wittgenstein’s Tractatus. From the Tractatus’s Picture theory of language we can strictly infer the two contradictory propositions simultaneously: the world and the language are equinumerous; the world and the language are not equinumerous. I call this antinomy the world-language paradox. (...) Based on a rigorous analysis of the Tractatus, with the help of the technical resources of Cantor’s naive set theory and Zermelo-Fraenkel set theory with the axiom of choice, I outline the world-language paradox and assess the unique possible solution plan, i.e., the mathematical plan utilizing ‘infinity’. I conclude that Wittgenstein cannot solve the hidden set-theoretical paradox of the Tractatus successfully unless he gives up his finitism. (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.

We present an order-theoretic analysis of set-theoretic paradoxes. This analysis will show that a large variety of purely set-theoretic paradoxes (including the various Russell paradoxes as well as all the familiar implementations of the paradoxes of Mirimanoff and Burali-Forti) are all instances of a single limitative phenomenon.

Georg Cantor's absolute infinity, the paradoxical Burali-Forti class Ω of all ordinals, is a monstrous non-entity for which being called a "class" is an undeserved dignity. This must be the ultimate vexation for mathematical philosophers who hold on to some residual sense of realism in set theory. By careful use of Ω, we can rescue Georg Cantor's 1899 "proof" sketch of the Well-Ordering Theorem––being generous, considering his declining health. We take the contrapositive of Cantor's suggestion and add Zermelo's (...) choice function. This results in a concise and uncomplicated proof of the Well-Ordering Theorem. (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)

I argue that absolutism, the view that absolutely unrestricted quantification is possible, is to blame for both the paradoxes that arise in naive set theory and variants of these paradoxes that arise in plural logic and in semantics. The solution is restrictivism, the view that absolutely unrestricted quantification is not possible. -/- It is generally thought that absolutism is true and that restrictivism is not only false, but inexpressible. As a result, the paradoxes are blamed, not on illicit quantification, but (...) on the logical conception of set which motivates naive set theory. The accepted solution is to replace this with the iterative conception of set. -/- I show that this picture is doubly mistaken. After a close examination of the paradoxes in chapters 2--3, I argue in chapters 4 and 5 that it is possible to rescue naive set theory by restricting quantification over sets and that the resulting restrictivist set theory is expressible. In chapters 6 and 7, I argue that it is the iterative conception of set and the thesis of absolutism that should be rejected. (shrink)

This work is, in large part, a series of refutations; it is also the author's Ph.D. thesis. First to be refuted is Russell's vicious circle principle as a general remedy for the solution of the paradoxes. The author rejects the classification of paradoxes into syntactic and semantic, since in his view there are no purely syntactic paradoxes. The distinction in logic between the uninterpreted syntactical aspect of a system and the system when given a determinate interpretation is held to be (...) untenable. Tarski's distinction between object-language and meta-language and his concept of semantically closed language are considered irrelevant for the solution of the Liar paradox. The author claims that the usual versions of the Liar paradox have the same structure as the Barber paradox, viz., [S ↔ ~S]. The author solves the Liar paradox by pointing out that it does not have a proper reference. Cantor's diagonal argument for the indenumerability [[sic]] of the real numbers is labeled as unsatisfactory. Since the diagonal number is dependent upon the real numbers in the constructed list, the author claims that this makes the diagonal number to be of a different nature and status than the real numbers in the list; thus we have what the author calls the dependence fallacy. The author also refuses to accept Cantor's nested interval proof of the indenumerability [[sic]] of the real numbers. Within the proof, two infinite sequences are constructed, each of which converges to a limit. Because the proof does not give a "definite rule of convergence," the author is not satisfied that the infinite sequences converge. Also rejected is Cantor's theorem. Other paradoxes analyzed are the Berry, Richard, heterological, "Richardian", Russell, Cantor, and Burali-Forti paradoxes.--T. G. N. (shrink)

The injective version of Cantor’s theorem appears in full second-order logic as the inconsistency of the abstraction principle, Frege’s Basic Law V (BLV), an inconsistency easily shown using Russell’s paradox. This incompatibility is akin to others—most notably that of a (Dedekind) infinite universe with the Nuisance Principle (NP) discussed by neo-Fregean philosophers of mathematics. This paper uses the Burali–Forti paradox to demonstrate this incompatibility, and another closely related, without appeal to principles related to the axiom of (...) choice—a result hitherto unestablished. It discusses both the general interest of this result, its interest to neo-Fregean philosophy of mathematics, and the potential significance of the Burali–Fortian method of proof. (shrink)

A number of authors have recently weighed in on the issue of whether it is coherent to have bound variables that range over absolutely everything. Prima facie, it is difficult, and perhaps impossible, to coherently state the “relativist” position without violating it. For example, the relativist might say, or try to say, that for any quantifier used in a proposition of English, there is something outside of its range. What is the range of this quantifier? Or suppose we ask the (...) relativist if there are some things that cannot appear in the range of any bound variable. The likely response would be along these lines: “No. For each object o, it possible to include o in the range of quantifiers, but one cannot quantify over everything at once.” This sentence contains unrestricted quantifiers, or so it seems, pending some clever move from a relativist. On the other hand, in the context of set theory, the reasoning behind the Burali-Forti paradox strongly suggests that there are well-orderings strictly longer than the collection of all ordinals. And set theorists regularly do transfinite recursions and transfinite reductions along such well-orderings. The relativist simply points out that one can always define new ordinals, and thus expand the range of one’s bound variables. The purpose of this paper is to explore the iterative framework, proposed in Zermelo’s 1930 paper, “Über Grenzzahlen und Mengenbereiche” (“On boundary numbers and domains of sets”), in order to shed light on these issues, and see what is involved in resolving them. (shrink)

Tento článek se zabývá dvěma aritmetickými systémy - konkrétně systémem, který představil R. Dedekind a systémem, který vytvořil G. Frege - a paradoxy, které se zde vyskytují - tedy Burali-Fortiho paradoxem (což je vůbec první fomrulace moderního paradoxu), Cantorovým paradoxem a Russellovým paradoxem. Hlavním cílem je ukázat, co mají tyto paradoxy společného a zdůvodnit, že ačkoli se tyto paradoxy vyskytují v různých systémech, mají společné znaky. Na základě studia uvedených systémů, paradoxů i různých řešení těchto paradoxů, autorka dospívá k (...) tvrzení, že zkoumané paradoxy vznikají na stejném základě, a to na základě problému nehierarchizovanosti.V dodatku tohoto článku navíc čtenář nalezne dva zajímavé historické exkurzy. První z nich předkládá v historicky autentické podobě odvození Burali-Fortiho paradoxu Druhý exkurz je věnován Russellovu paradoxu a představuje jej nejen v podobě, jak jej vyjádřil B. Russell, ale také ve formulaci G. Frega. (shrink)

Tento článek se zabývá dvěma aritmetickými systémy - konkrétně systémem, který představil R. Dedekind a systémem, který vytvořil G. Frege - a paradoxy, které se zde vyskytují - tedy Burali-Fortiho paradoxem (což je vůbec první fomrulace moderního paradoxu), Cantorovým paradoxem a Russellovým paradoxem. Hlavním cílem je ukázat, co mají tyto paradoxy společného a zdůvodnit, že ačkoli se tyto paradoxy vyskytují v různých systémech, mají společné znaky. Na základě studia uvedených systémů, paradoxů i různých řešení těchto paradoxů, autorka dospívá k (...) tvrzení, že zkoumané paradoxy vznikají na stejném základě, a to na základě problému nehierarchizovanosti.V dodatku tohoto článku navíc čtenář nalezne dva zajímavé historické exkurzy. První z nich předkládá v historicky autentické podobě odvození Burali-Fortiho paradoxu Druhý exkurz je věnován Russellovu paradoxu a představuje jej nejen v podobě, jak jej vyjádřil B. Russell, ale také ve formulaci G. Frega. (shrink)

Such contradictions arise “at the limits of thought” in the following sense: we have reason to set boundaries to certain conceptual processes, which, however, turn out to actually cross those boundaries. The boundaries cannot be crossed, yet they can, for they are crossed. For example, Kant regarded noumena as beyond the limit of the conceivable, yet he made judgments about them, so he did conceive of them. For another example, Russell’s theory of types cannot be expressed, yet he does express (...) it. And so on, from Aristotle’s notion of prime matter to Derrida’s différance. The boundaries that cannot be but are crossed may concern iteration, expression, cognition, or conception. In most cases, a single argument pattern is operative, according to Priest. He calls it the Inclosure Schema [=IS]. It is a contradiction-generating mechanism that works as follows: suppose we define a set Ω, on the basis of a condition φ ); suppose that Ω exists and that it has property ψ. Next, suppose we can define a function δ such that, for any subset x of Ω that has property ψ, we have both δ ∉ x and δ ∈ Ω. As Ω is a subset of itself and it has ψ by hypothesis, a contradiction follows: both δ ∈ Ω and δ ∉ Ω. The two sides of the contradiction—or perhaps the operations by which they are established—are called Closure and Transcendence. ). For example, take Burali Forti’s paradox that is greater than all members of On, and therefore not an ordinal). Here Ω = On, and δ is the function that assigns to x the least ordinal greater than all members of x is both an ordinal—δ ∈ Ω, “Closure”—and not a member of x ). The condition φ is just the property of being an ordinal. (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)

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)

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)

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)

Enmarcado dentro del tema principal del pensamiento del futuro, la muerte heideggeriana y la redención de Rosenzweig presentan dos formas diferentes de apuntar a aspectos que pueden arrojar luz sobre la existencia humana. Más en concreto, el modo en que Heidegger y Rosenzweig conciben el futuro se encuentra en las raíces de cuestiones tales como la mortalidad, la otredad y la posibilidad.

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)

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.