The present note was prompted by Weber’s approach to proving Cantor’s theorem, i.e., the claim that the cardinality of the power set of a set is always greater than that of the set itself. While I do not contest that his proof succeeds, my point is that he neglects the possibility that by similar methods it can be shown also that no non-empty set satisfies Cantor’s theorem. In this paper unrestricted abstraction based on a cut free (...) Gentzen type sequential calculus will be employed to prove both results. In view of the connection between Priest’s three-valued logic of paradox and cut free Gentzen calculi this, a fortiori, has an impact on any paraconsistent set theory built on Priest’s logic of paradox. (shrink)

An intuitionistic version of Cantor's theorem, which shows that there is no surjective function from the type of the natural numbers N into the type N → N of the functions from N into N, is proved within Martin-Löf's Intuitionistic Type Theory with the universe of the small types.

Although recent evidence is somewhat ambiguous, if not confusing, Patrick Grim still seems to believe that his Cantorian argument against omniscienceis sound. According to this argument, it follows by Cantor’s power set theorem that there can be no set of all truths. Hence, assuming that omniscience presupposes precisely such a set, there can be no omniscient being. Reconsidering this argument, however, guided in particular by Alvin Plantinga’s critique thereof, I find it far from convincing. Not only does it (...) have an enormously untoward side effect, but it is self-referentially incoherent as well. (shrink)

In these articles, I describe Cantor’s power-class theorem, as well as a number of logical and philosophical paradoxes that stem from it, many of which were discovered or considered (implicitly or explicitly) in Bertrand Russell’s work. These include Russell’s paradox of the class of all classes not members of themselves, as well as others involving properties, propositions, descriptive senses, class-intensions, and equivalence classes of coextensional properties. Part I focuses on Cantor’s theorem, its proof, how it can (...) be used to manufacture paradoxes, Frege’s diagnosis of the core difficulty, and several broad categories of strategies for offering solutions to these paradoxes. (shrink)

Sequel to Part I. In these articles, I describe Cantor’s power-class theorem, as well as a number of logical and philosophical paradoxes that stem from it, many of which were discovered or considered (implicitly or explicitly) in Bertrand Russell’s work. These include Russell’s paradox of the class of all classes not members of themselves, as well as others involving properties, propositions, descriptive senses, class-intensions and equivalence classes of coextensional properties. Part II addresses Russell’s own various attempts to solve (...) these paradoxes, including strategies that he considered and rejected (limitation of size, the zigzag theory, etc.), as well as his own final views whereupon many purported entities that, if reified, lead to these contradictions, must not be genuine entities, but ‘logical fictions’ or ‘logical constructions’ instead. (shrink)

It is claimed that cantor had the technical apparatus available to derive russell's paradox some ten years before russell's discovery.

Although recent evidence is somewhat ambiguous, if not confusing, Patrick Grim still seems to believe that his Cantorian argument against omniscienceis sound. According to this argument, it follows by Cantor’s power set theorem that there can be no set of all truths. Hence, assuming that omniscience presupposes precisely such a set, there can be no omniscient being. Reconsidering this argument, however, guided in particular by Alvin Plantinga’s critique thereof, I find it far from convincing. Not only does it (...) have an enormously untoward side effect, but it is self-referentially incoherent as well. (shrink)

I argue that Composition as Identity blocks the plural version of Cantor's Theorem, and that therefore the plural version of Cantor's Theorem can no longer be uncritically appealed to. As an example, I show how this result blocks a recent argument by Hawthorne and Uzquiano.

The following essay reconsiders the ontological and logical issues around Frege’s Basic Law (V). If focuses less on Russell’s Paradox, as most treatments of Frege’s Grundgesetze der Arithmetik (GGA)1 do, but rather on the relation between Frege’s Basic Law (V) and Cantor’s Theorem (CT). So for the most part the inconsistency of Naïve Comprehension (in the context of standard Second Order Logic) will not concern us, but rather the ontological issues central to the conflict between (BLV) and (CT). (...) These ontological issues are interesting in their own right. And if and only if in case ontological considerations make a strong case for something like (BLV) we have to trouble us with inconsistency and paraconsistency. These ontological issues also lead to a renewed methodological reflection what to assume or recognize as an axiom. (shrink)

We construct a novel proof of Cantor's theorem in set theory.

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)

George Boolos was one of the most prominent and influential logician-philosophers of recent times. This collection, nearly all chosen by Boolos himself shortly before his death, includes thirty papers on set theory, second-order logic, and plural quantifiers; on Frege, Dedekind, Cantor, and Russell; and on miscellaneous topics in logic and proof theory, including three papers on various aspects of the Gödel theorems. Boolos is universally recognized as the leader in the renewed interest in studies of Frege's work on logic and (...) the philosophy of mathematics. John Burgess has provided introductions to each of the three parts of the volume, and also an afterword on Boolos's technical work in provability logic, which is beyond the scope of this volume. (shrink)

Cantor’s proof that the powerset of the set of all natural numbers is uncountable yields a version of Richard’s paradox when restricted to the full definable universe, that is, to the universe containing all objects that can be defined not just in one formal language but by means of the full expressive power of natural language: this universe seems to be countable on one account and uncountable on another. We argue that the claim that definitional contexts impose restrictions on (...) the scope of quantifiers reveals a natural way out. (shrink)

We first consider the entailment logic MC, based on meaning containment, which contains neither the Law of Excluded Middle (LEM) nor the Disjunctive Syllogism (DS). We then argue that the DS may be assumed at least on a similar basis as the assumption of the LEM, which is then justified over a finite domain or for a recursive property over an infinite domain. In the latter case, use is made of Mathematical Induction. We then show that an instance of the (...) LEM is intrumental in the proof of Cantor's Theorem, and we then argue that this is based on a more general form than can be reasonably justified. We briefly consider the impact of our approach on arithmetic and naive set theory, and compare it with intuitionist mathematics and briefly with recursive mathematics. Our "Four Basic Logical Issues" paper would provide useful background, the current paper being an application of the some of the ideas in it. (shrink)

The paper claims that the strategy adopted in the proof of Cantor’s theorem is problematic. Using the strategy, an unacceptable situation is built. The paper also makes the suggestion that the proof of Cantor’s theorem is possible due to lack of an apparatus to represent emptiness at a certain level in the ontology of set-theory.

Two thoughts about the concept of number are incompatible: that any zero or more things have a number, and that any zero or more things have a number only if they are the members of some one set. It is Russell's paradox that shows the thoughts incompatible: the sets that are not members of themselves cannot be the members of any one set. The thought that any things have a number is Frege's; the thought that things have a number only (...) if they are the members of a set may be Cantor's and is in any case a commonplace of the usual contemporary presentations of the set theory that originated with Cantor and has become ZFC.In recent years a number of authors have examined Frege's accounts of arithmetic with a view to extracting an interesting subtheory from Frege's formal system, whose inconsistency, as is well known, was demonstrated by Russell. These accounts are contained in Frege's formal treatise Grundgesetze der Arithmetik and his earlier exoteric book Die Grundlagen der Arithmetik. We may describe the two central results of the recent re-evaluation of his work in the following way: Let Frege arithmetic be the result of adjoining to full axiomatic second-order logic a suitable formalization of the statement that the Fs and the Gs have the same number if and only if the F sand the Gs are equinumerous. (shrink)

Two thoughts about the concept of number are incompatible: that any zero or more things have a (cardinal) number, and that any zero or more things have a number (if and) only if they are the members of some one set. It is Russell's paradox that shows the thoughts incompatible: the sets that are not members of themselves cannot be the members of any one set. The thought that any (zero or more) things have a number is Frege's; the thought (...) that things have a number only if they are the members of a set may be Cantor's and is in any case a commonplace of the usual contemporary presentations of the set theory that originated with Cantor and has become ZFC. In recent years a number of authors have examined Frege's accounts of arithmetic with a view to extracting an interesting subtheory from Frege's formal system, whose inconsistency, as is well known, was demonstrated by Russell. These accounts are contained in Frege's formal treatise Grundgesetze der Arithmetik and his earlier exoteric book Die Grundlagen der Arithmetik. We may describe the two central results of the recent re-evaluation of his work in the following way: Let Frege arithmetic be the result of adjoining to full axiomatic second-order logic a suitable formalization of the statement that the Fs and the Gs have the same number if and only if the F sand the Gs are equinumerous. (shrink)

This invaluable resource is the first one-volume, in-depth, comprehensive history of modern science ever published.

Critics of South Park—and they are legion—bitterly complain about its relentless obscenity and potty humor. But if one wanted to mount a high‐minded defense of the show's low‐minded jokes, one might go all the way back to Plato to find a link between philosophy and vulgarity. Cartman fights the countercultural forces who invade South Park and mindlessly blames all the troubles of America on “the corporations.” In “Gnomes” a national coffee chain called Harbucks—an obvious reference to Starbucks—comes to South Park (...) and tries to buy out the local Tweek Bros. coffee shop. Having had the audacity to defend Starbucks, in its eighth season South Park went on to rally to the cause of Wal‐Mart—under an even more thinly disguised name in an episode called “Something Wall Mart This Way Comes.” Once again, South Park proclaims the sovereignty of the consumer in a market economy. (shrink)

It is proved in $\mathsf {ZF}$ (without the axiom of choice) that, for all infinite sets M, there are no surjections from $\omega \times M$ onto $\operatorname {\mathrm {\mathscr {P}}}(M)$.

Chapter Seven discusses Edgar Ulmer's The Black Cat as the creation of a high modernist European émigré working in a lowbrow American genre, the horror movie. Ulmer portrays a self-destructive generation of Europeans permanently scarred by the horrors of World War I. Into their world, he introduces a young American couple on their honeymoon, who, in their naïveté, are almost destroyed by the mad Europeans, but escape their satanic clutches in the end. Hoping to succeed in his newly adopted homeland, (...) Ulmer seems to turn his back on his European heritage as a dead end, and yet he cannot help suggesting the cultural superiority of his sophisticated European characters. The Black Cat takes its place in a long line of European works critical of American culture, while at the same time raising serious doubts about Europe's future in the 1930s. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Uncanonical Dante: The Divine Comedy And Islamic PhilosophyPaul A. CantorThe distorted notions of invisible things which Dante and hisrival Milton have idealized, are merely the mask and the mantlein which these great poets walk through eternity enveloped anddisguised. It is a difficult question to determine how far theywere conscious of the distinction which must have subsisted intheir minds between their own creeds and that of the people.Dante at (...) least appears to wish to mark the full extent of it byplacing Riphaeus, whom Virgil calls justissimus unus, inParadise, and observing a most heretical caprice in hisdistribution of rewards and punishments.—Percy Shelley, A Defence of PoetryIThe case of Dante provides an excellent opportunity to open up the question of the Western canon. In one sense, Dante is the perfect example of a canonical author. His name is one of the few certain to appear on anybody’s short list of the truly central authors in the Western literary tradition. But in another sense Dante can be regarded as uncanonical. In his own day he was widely suspected of being heretical in his religious views, 1 and a careful reading of his works does indeed raise serious doubts about his being the pillar of orthodoxy he is often taken to be today. 2 Out of this interplay between the canonical and the noncanonical Dante, I hope to show that the issue of the Western canon is more complicated than either its defenders or its attackers generally present it.In discussing the issue of the canon, it is important to sort out at the [End Page 138] beginning what we do and do not mean by the term. A canonical work may merely be a work that has been accepted into the literary canon, one that has become a touchstone in the reading and teaching of literature. But the term canonical can suggest something else, that the work is orthodox and somehow represents a central authoritative position in Western culture. The word canonical is so loaded with religious connotations that it is difficult to separate the relatively neutral first meaning of the term from the loaded second meaning. Dante is a case in point. When people refer to him as a canonical author, they usually do not simply mean that he is widely read and taught. Most discussions of Dante today treat him as representing an authoritative cultural moment in the Western tradition, as the supreme embodiment of the medieval mind. Viewed that way, Dante becomes an emblem of everything contemporary critics of the Western canon bitterly hate and reject. The reason they feel that they must attack authors like Dante and displace them from the center of literary study is that these authors have come to stand for orthodoxy and thus seem to enforce the hegemony of Western culture.Critics who wish to champion various forms of non-Western culture have a particular axe to grind with canonical authors like Dante. The contemporary debate over the Western canon seems to be premised on a sharp opposition between Western and non-Western cultures, as if they were complete and irreconcilable antitheses, and even wholly unrelated. One of the principal charges against the Western canon is that it is Eurocentric, that it remains confined within a narrow orbit of European ideas and beliefs, thus excluding all other views of the world. A corrolary of the idea of Eurocentrism is the concept of Orientalism, developed by Edward Said. 3 Said argues that throughout its history, the Occident has defined itself in opposition to the Orient, basing its elevated self-image on a debased vision of the cultural Other. In Said’s argument, the Occident views itself as rational as opposed to an irrational Orient, as emotionally disciplined in contrast to an emotionally uncontrolled Orient, and as masculine over against a feminine Orient.In medieval Europe the Orient was chiefly represented by the Muslim world, and one does not have to look far in medieval literature to find the kinds of orientalist stereotypes about which Said writes. The French Song of Roland contains excellent examples, but even the Divine Comedy seems to... (shrink)

In 1956, a few writings of Wittgenstein that he didn't publish in his lifetime were revealed to the public. These writings were gathered in the book Remarks on the Foundations of Mathematics (1956). There, we can see that Wittgenstein had some discontentment with the way philosophers, logicians, and mathematicians were thinking about paradoxes, and he even registered a few polemic reasons to not accept Gödel’s incompleteness theorems.

In this paper several problems are raised concerning Ian Barbour's four ways of interrelating science and religion—Conflict, Independence, Dialogue, and Integration—as put forward in such publications as his highly influential Religion in an Age of Science (1990) and widely adopted by other writers in this field. The authors argue that this taxonomy is not very useful or analytically helpful, especially to historians seeking to understand past engagements between science and religion.

This collection of essays, offered in honor of the distinguished career of prominent political philosophy professor Clifford Orwin, brings together internationally renowned scholars to provide a wide context and discuss various aspects of the virtue of “humanity” through the history of political philosophy.

This collection of essays, offered in honor of the distinguished career of prominent political philosophy professor Clifford Orwin, brings together internationally renowned scholars to provide a wide context and discuss various aspects of the virtue of “humanity” through the history of political philosophy.

The “happy fish” passage concluding the “Autumn Floods” chapter of the Classical Chinese text known as the Zhuangzi has traditionally been seen to advance a form of relativism which precludes objectivity. My aim in this paper is to question this view with close reference to the passage itself. I further argue that the central concern of the two philosophical personae in the passage – Zhuangzi and Huizi – is not with the epistemic standards of human judgements (the established view since (...) Hansen, “The Relatively Happy Fish”), but with the more basic problem of species-specific perspectives. On my reading, Zhuangzi’s emphatic positionality in the passage – on the dam, accompanied by his friend Huizi – plausibly suggests a circumspect reflection on the limitedness of human knowledge. It is significant that Zhuangzi’s knowledge of fish happiness is avowedly from a certain place, and not absolute. But there is still a sense in which this view is objective: namely, insofar as it adequately accounts for an inherently human perspective on the world. I call this modest form of relativism ‘Species Relativism’, which, crucially, leaves room for objectivity, even though a fully objective (i.e. absolute) view of the world is not accessible to humans. (shrink)

Some people will confront Alzheimer's with a measure of resignation, a determination to struggle against the progressive debilitation and to extract whatever comforts and benefits they can from their remaining existence. They are entitled to pursue that resolute path. For other people, like myself, protracted maintenance during progressive cognitive dysfunction and helplessness is an intolerably degrading prospect. The critical question for those of us seeking to avoid protracted dementia is how best to accomplish that objective.One strategy is to engineer one's (...) own death while still mentally competent to do so (even in the stage of mild dementia). If I were to use a preemptive strategy in the face of a dementia diagnosis, I would probably choose to stop eating and drinking, a process known as voluntarily stopping eating and drinking. An alternative tactic for avoiding prolonged dementia would be to allow oneself to decline into moderate dementia—thus losing capacity to perform self‐deliverance or even to make serious medical decisions—but before getting to that point to provide advance instructions rejecting prospective life‐sustaining medical interventions. These advance instructions would authorize palliative but not curative measures. My current personal instructions define the point of intolerable cognitive decline triggering medical nonintervention as “mental deterioration to a point when I can no longer read and understand written material such as a newspaper or financial records such as a checkbook.” These instructions dictate allowing my demise at a point of moderate dementia when I may not be perceptibly suffering, when I may still be getting some rudimentary satisfaction from my debilitated life, and when I no longer recall the preoccupation with personally intolerable indignity that motivated my instructions. Can I expect that my advance instructions will be implemented in those circumstances? Is it lawful, and is it moral for a surrogate decision‐maker and associated caregivers to allow an uncomprehending, ostensibly content but demented individual to die? My analysis herein contends that it is not only lawful and moral but also legally required to implement clear, considered advance instructions even at a stage of moderate dementia. (shrink)

While many aspects of Shapin's historical thesis are accepted, this paper raises objections to specific parts of his historical account, and also to the historiographical assumptions underlying his sociological programme. In particular, Shapin's claim to have explained the Edinburgh phrenology debate in social terms is analysed and rejected.