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Set Theorists 1

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Set theorists From Wikipedia, the free encyclopedia
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Set theoristsFrom Wikipedia, the free encyclopedia

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Chapter 1

Alain Badiou

Alain Badiou (French: [alɛ̃ badju] (listen) ; born 17 January 1937) is a French philosopher, formerly chair ofPhilosophy at the École Normale Supérieure (ENS) and founder of the faculty of Philosophy of the Université deParis VIII with Gilles Deleuze, Michel Foucault and Jean-François Lyotard. Badiou has written about the concepts ofbeing, truth, event and the subject in a way that, he claims, is neither postmodern nor simply a repetition of modernity.Badiou has been involved in a number of political organisations, and regularly comments on political events. Badiouargues for resurrecting the idea of communism.

1.1 Biography

Badiou is the son of mathematician and member of the Resistance in France during World War II Raymond Ba-diou (1905–1996). He was a student at the Lycée Louis-Le-Grand and then the École Normale Supérieure (1955–1960).[1] In 1960, he wrote his diplôme d'études supérieures (roughly equivalent to an MA thesis) on Spinoza forGeorges Canguilhem (the topic was “Demonstrative Structures in the First Two Books of Spinoza’s Ethics", "Struc-tures démonstratives dans les deux premiers livres de l'Éthique de Spinoza").[2] He taught at the lycée in Reims from1963 where he became a close friend of fellow playwright (and philosopher) François Regnault,[3] and published acouple of novels before moving first to the faculty of letters of the University of Reims (the collège littéraire univer-sitaire)[4] and then to the University of Paris VIII (Vincennes-Saint Denis) in 1969.[5] Badiou was politically activevery early on, and was one of the founding members of the Unified Socialist Party (PSU). The PSU was particularlyactive in the struggle for the decolonization of Algeria. He wrote his first novel, Almagestes, in 1964. In 1967 hejoined a study group organized by Louis Althusser, became increasingly influenced by Jacques Lacan and became amember of the editorial board of Cahiers pour l'Analyse.[5] By then he “already had a solid grounding in mathematicsand logic (along with Lacanian theory)",[5] and his own two contributions to the pages of Cahiers “anticipate manyof the distinctive concerns of his later philosophy”.[5]

The student uprisings ofMay 1968 reinforced Badiou’s commitment to the far Left, and he participated in increasinglymilitant groups, such as the Union des communistes de France marxiste-léniniste (UCFml). To quote Badiou himself,the UCFml is “the Maoist organization established in late 1969 by Natacha Michel, Sylvain Lazarus, myself and afair number of young people”.[6] During this time, Badiou joined the faculty of the newly founded University of ParisVIII/Vincennes-Saint Denis which was a bastion of counter-cultural thought. There he engaged in fierce intellectualdebates with fellow professors Gilles Deleuze and Jean-François Lyotard, whose philosophical works he consideredunhealthy deviations from the Althusserian program of a scientific Marxism.In the 1980s, as both Althusserian Marxism and Lacanian psychoanalysis went into decline (after Lacan died andAlthusser was committed to a psychiatric hospital), Badiou publishedmore technical and abstract philosophical works,such as Théorie du sujet (1982), and his magnum opus, Being and Event (1988). Nonetheless, Badiou has neverrenounced Althusser or Lacan, and sympathetic references to Marxism and psychoanalysis are not uncommon in hismore recent works (most notably Petit panthéon portatif/Pocket Pantheon).[7][8]

He took up his current position at the ENS in 1999. He is also associated with a number of other institutions, such asthe Collège International de Philosophie. He was a member of “L'Organisation Politique” which, as mentioned above,he founded in 1985 with some comrades from the Maoist UCFml. This organization disbanded in 2007, accordingto the French Wikipedia article (linked to in the previous sentence). In 2002, he was a co-founder of the Centre


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International d'Etude de la Philosophie Française Contemporaine, alongside Yves Duroux and his former studentQuentin Meillassoux.[9] Badiou has also enjoyed success as a dramatist with plays such as Ahmed le Subtil.In the last decade, an increasing number of Badiou’s works have been translated into English, such as Ethics, Deleuze,Manifesto for Philosophy, Metapolitics, and Being and Event. Short pieces by Badiou have likewise appeared inAmerican and English periodicals, such as Lacanian Ink, New Left Review, Radical Philosophy, Cosmos and Historyand Parrhesia. Unusually for a contemporary European philosopher his work is increasingly being taken up bymilitants in countries like India, the Democratic Republic of Congo and South Africa.Recently Badiou got into a fierce controversy within the confines of Parisian intellectual life. It started in 2005 with thepublication of his “Circonstances 3: Portées du mot 'juif'" – The Uses of the Word “Jew”.[10] This book generateda strong response with calls of Badiou being labelled Anti-Semitic. The wrangling became a cause célèbre witharticles going back and forth in the French newspaper Le Monde and in the cultural journal “Les temps modernes.”Linguist and Lacanian philosopher Jean-Claude Milner, a past president of Jacques Derrida's Collège internationalde philosophie, has accused Badiou of Anti-Semitism.[11]

In 2014–15, Badiou had the role of Honorary President at The Global Center for Advanced Studies.[12]

1.2 Key concepts

Badiou makes repeated use of several concepts throughout his philosophy. One of the aims of his thought is to showthat his categories of truth are useful for any type of philosophical critique. Therefore, he uses them to interrogateart and history as well as ontology and scientific discovery. Johannes Thumfart argues that Badiou’s philosophy canbe regarded as a contemporary reinterpretation of Platonism.[13]

1.2.1 Conditions

According to Badiou, philosophy is suspended from four conditions (art, love, politics, and science), each of them fullyindependent “truth procedures.” (For Badiou’s notion of truth procedures, see below.) Badiou consistently maintainsthroughout his work (but most systematically inManifesto for Philosophy) that philosophy must avoid the temptationto suture itself (that is, to hand over its entire intellectual effort) to any of these independent truth procedures. Whenphilosophy does suture itself to one of its conditions (and Badiou argues that the history of philosophy during thenineteenth and twentieth centuries is primarily a history of sutures), what results is a philosophical “disaster.” Conse-quently, philosophy is, according to Badiou, a thinking of the compossibility of the several truth procedures, whetherthis is undertaken through the investigation of the intersections between distinct truth procedures (the intersectionof art and love in the novel, for instance), or whether this is undertaken through the more traditionally philosophicalwork of addressing categories like truth or the subject (concepts that are, as concepts, external to the individual truthprocedures, though they are functionally operative in the truth procedures themselves). For Badiou, when philosophyaddresses the four truth procedures in a genuinely philosophical manner, rather than through a suturing abandon-ment of philosophy as such, it speaks of them with a theoretical terminology that marks its philosophical character:“inaesthetics” rather than art; metapolitics rather than politics; ontology rather than science; etc.Truth, for Badiou, is a specifically philosophical category. While philosophy’s several conditions are, on their ownterms, “truth procedures” (i.e., they produce truths as they are pursued), it is only philosophy that can speak of theseveral truth procedures as truth procedures. (The lover, for instance, does not think of her love as a question oftruth, but simply and rightly as a question of love. Only the philosopher sees in the true lover’s love the unfolding ofa truth.) Badiou has a very rigorous notion of truth, one that is strongly against the grain of much of contemporaryEuropean thought. Badiou at once embraces the traditional modernist notion that truths are genuinely invariant(always and everywhere the case, eternal and unchanging) and the incisively postmodernist notion that truths areconstructed through processes. Badiou’s theory of truth, exposited throughout his work, accomplishes this strangemixture by uncoupling invariance from self-evidence (such that invariance does not imply self-evidence), as well asby uncoupling constructedness from relativity (such that constructedness does not lead to relativism).The idea, here, is that a truth’s invariance makes it genuinely indiscernible: because a truth is everywhere and alwaysthe case, it passes unnoticed unless there is a rupture in the laws of being and appearance, during which the truthin question becomes, but only for a passing moment, discernible. Such a rupture is what Badiou calls an event,according to a theory originally worked out in Being and Event and fleshed out in important ways in Logics of Worlds.The individual who chances to witness such an event, if he is faithful to what he has glimpsed, can then introducethe truth by naming it into worldly situations. For Badiou, it is by positioning oneself to the truth of an event that a

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human animal becomes a subject; subjectivity is not an inherent human trait. According to a process or procedurethat subsequently unfolds only if those who subject themselves to the glimpsed truth continue to be faithful in thework of announcing the truth in question, genuine knowledge is produced (knowledge often appears in Badiou’s workunder the title of the “veridical”). While such knowledge is produced in the process of being faithful to a truth event,it should be noted that, for Badiou, knowledge, in the figure of the encyclopedia, always remains fragile, subject towhat may yet be produced as faithful subjects of the event produce further knowledge. According to Badiou, truthprocedures proceed to infinity, such that faith (fidelity) outstrips knowledge. (Badiou, following both Lacan andHeidegger, distances truth from knowledge.) The dominating ideology of the day, which Badiou terms “democraticmaterialism,” denies the existence of truth and only recognizes "bodies" and "languages.” Badiou proposes a turntowards the "materialist dialectic,” which recognizes that there are only bodies and languages, except there are alsotruths.

1.2.2 Inaesthetic

In Handbook of Inaesthetics Badiou both draws on the original Greek meaning and the later Kantian concept of“aesthesis” as “material perception” and coins the phrase “inaesthetic” to refer to a concept of artistic creation thatdenies “the reflection/object relation” yet, at the same time, in reaction against the bourgeois idea of mimesis, orpoetic reflection of “nature”, he affirms that art is “immanent” and “singular”. Art is immanent in the sense that itstruth is given in its immediacy in a given work of art, and singular in that its truth is found in art and art alone—hencereviving the ancient materialist concept of “aesthesis”. His view of the link between philosophy and art is tied intothe motif of pedagogy, which he claims functions so as to “arrange the forms of knowledge in a way that some truthmay come to pierce a hole in them”. He develops these ideas with examples from the prose of Samuel Beckett andthe poetry of Stéphane Mallarmé and Fernando Pessoa (who he argues has developed a body of work that philosophyis currently incapable of incorporating), among others.

1.3 Introduction to Being and Event

The major propositions of Badiou’s philosophy all find their basis in Being and Event, in which he continues hisattempt (which he began in Théorie du sujet) to reconcile a notion of the subject with ontology, and in particular post-structuralist and constructivist ontologies.[14] A frequent criticism of post-structuralist work is that it prohibits, throughits fixation on semiotics and language, any notion of a subject. Badiou’s work is, by his own admission,[15] an attemptto break out of contemporary philosophy’s fixation upon language, which he sees almost as a straitjacket. This effortleads him, inBeing and Event, to combine rigorousmathematical formulae with his readings of poets such asMallarméand Hölderlin and religious thinkers such as Pascal. His philosophy draws upon both 'analytical' and 'continental'traditions. In Badiou’s own opinion, this combination places him awkwardly relative to his contemporaries, meaningthat his work had been only slowly taken up.[16] Being and Event offers an example of this slow uptake, in fact: it wastranslated into English only in 2005, a full seventeen years after its French publication.As is implied in the title of the book, two elements mark the thesis of Being and Event: the place of ontology, or'the science of being qua being' (being in itself), and the place of the event – which is seen as a rupture in being– through which the subject finds realization and reconciliation with truth. This situation of being and the rupturewhich characterizes the event are thought in terms of set theory, and specifically Zermelo–Fraenkel set theory (withthe axiom of choice), to which Badiou accords a fundamental role in a manner quite distinct from the majority ofeither mathematicians or philosophers.

1.3.1 Mathematics as ontology

For Badiou the problem which the Greek tradition of philosophy has faced and never satisfactorily dealt with is thatwhile beings themselves are plural, and thought in terms of multiplicity, being itself is thought to be singular; that is,it is thought in terms of the one. He proposes as the solution to this impasse the following declaration: that the one isnot. This is why Badiou accords set theory (the axioms of which he refers to as the Ideas of the multiple) such stature,and refers to mathematics as the very place of ontology: Only set theory allows one to conceive a 'pure doctrine ofthe multiple'. Set theory does not operate in terms of definite individual elements in groupings but only functionsinsofar as what belongs to a set is of the same relation as that set (that is, another set too). What individuates a set,therefore, is not an existential positive proposition, but other multiples whose properties (i.e., structural relations)validate its presentation. The structure of being thus secures the regime of the count-as-one. So if one is to think of

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a set – for instance, the set of people, or humanity – as counting as one, the multiple elements which belong to thatset are secured as one consistent concept (humanity), but only in terms of what does not belong to that set. Whatis crucial for Badiou is that the structural form of the count-as-one, which makes multiplicities thinkable, implies(somehow or other) that the proper name of being does not belong to an element as such (an original 'one'), but ratherthe void set (written Ø), the set to which nothing (not even the void set itself) belongs. It may help to understand theconcept 'count-as-one' if it is associated with the concept of 'terming': a multiple is not one, but it is referred to with'multiple': one word. To count a set as one is to mention that set. How the being of terms such as 'multiple' doesnot contradict the non-being of the one can be understood by considering the multiple nature of terminology: forthere to be a term without there also being a system of terminology, within which the difference between terms givescontext and meaning to any one term, is impossible. 'Terminology' implies precisely difference between terms (thusmultiplicity) as the condition for meaning. The idea of a term without meaning is incoherent, the count-as-one is astructural effect or a situational operation; it is not an event of 'truth'. Multiples which are 'composed' or 'consistent'are count-effects. 'Inconsistent multiplicity' [meaning?] is [somehow or other] 'the presentation of presentation.'Badiou’s use of set theory in this manner is not just illustrative or heuristic. Badiou uses the axioms of Zermelo–Fraenkel set theory to identify the relationship of being to history, Nature, the State, and God. Most significantlythis use means that (as with set theory) there is a strict prohibition on self-belonging; a set cannot contain or belongto itself. This results from the axiom of foundation – or the axiom of regularity – which enacts such a prohibition(cf. p. 190 in Being and Event). (This axiom states that every non-empty set A contains an element y that is disjointfrom A.) Badiou’s philosophy draws two major implications from this prohibition. Firstly, it secures the inexistenceof the 'one': there cannot be a grand overarching set, and thus it is fallacious to conceive of a grand cosmos, a wholeNature, or a Being of God. Badiou is therefore – against Georg Cantor, from whom he draws heavily – staunchlyatheist. However, secondly, this prohibition prompts him to introduce the event. Because, according to Badiou, theaxiom of foundation 'founds’ all sets in the void, it ties all being to the historico-social situation of the multiplicities ofde-centred sets – thereby effacing the positivity of subjective action, or an entirely 'new' occurrence. And whilst this isacceptable ontologically, it is unacceptable, Badiou holds, philosophically. Set theory mathematics has consequently'pragmatically abandoned' an area which philosophy cannot. And so, Badiou argues, there is therefore only onepossibility remaining: that ontology can say nothing about the event.Several critics have questioned Badiou’s use of mathematics. Mathematician Alan Sokal and physicist Jean Bricmontwrite that Badiou proposes, with seemingly “utter seriousness,” a blending of psychoanalysis, politics and set theorythat they contend is preposterous.[17] Similarly, philosopher Roger Scruton has questioned Badiou’s grasp of thefoundation of mathematics, writing in 2012:

There is no evidence that I can find in Being and Event that the author really understands what he istalking about when he invokes (as he constantly does) Georg Cantor’s theory of transfinite cardinals, theaxioms of set theory, Gödel’s incompleteness proof or Paul Cohen’s proof of the independence of thecontinuum hypothesis. When these things appear in Badiou’s texts it is always allusively, with fragmentsof symbolism detached from the context that endows them with sense, and often with free variables andbound variables colliding randomly. No proof is clearly stated or examined, and the jargon of set theoryis waved like a magician’s wand, to give authority to bursts of all but unintelligible metaphysics.[18]

An example of a critique from a mathematician’s point of view is the essay 'Badiou’s Number: A Critique of Math-ematics as Ontology' by Ricardo L. Nirenberg and David Nirenberg, which takes issue in particular with Badiou’smatheme of the Event in Being and Event, which has already been alluded to in respect of the 'axiom of foundation'above. Nirenberg and Nirenberg write:

Rather than being defined in terms of objects previously defined, ex is here defined in terms of itself;you must already have it in order to define it. Set theorists call this a not-well-founded set. This kindof set never appears in mathematics—not least because it produces an unmathematical mise-en-abîme:if we replace ex inside the bracket by its expression as a bracket, we can go on doing this forever—and so can hardly be called “a matheme.”' (http://criticalinquiry.uchicago.edu/uploads/pdf/nirenbergs_badiousnumber_complete.pdf, pp. 598–9)

1.3.2 The event and the subject

The principle of the event is where Badiou diverges from the majority of late twentieth century philosophy and socialthought, and in particular the likes of Foucault, Butler, Lacan and Deleuze, among others. In short, it represents that

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Drawing from 18 November 2006 “Truth procedure in politics” lecture

which is outside ontology. Badiou’s problem here is, unsurprisingly, the question of how to 'make use' of that whichcannot be discerned. But it is a problem he views as vital, because if one constructs the world only from that whichcan be discerned and therefore given a name, it results in either the destitution of subjectivity and the removal of thesubject from ontology (the criticism continually leveled at Foucault’s discursive universe), or the Panglossian solutionof Leibniz: that God is language in its supposed completion.Badiou again turns here to mathematics and set theory – Badiou’s language of ontology – to study the possibility of anindiscernible element existing extrinsically to the situation of ontology. He employs the strategy of the mathematicianPaul J. Cohen, using what are called the conditions of sets. These conditions are thought of in terms of domination, adomination being that which defines a set. (If one takes, in binary language, the set with the condition 'items markedonly with ones’, any item marked with zero negates the property of the set. The condition which has only ones is thusdominated by any condition which has zeros in it [cf. p. 367-71 in Being and Event].) Badiou reasons using theseconditions that every discernible (nameable or constructible) set is dominated by the conditions which don't possessthe property that makes it discernible as a set. (The property 'one' is always dominated by 'not one'.) These setsare, in line with constructible ontology, relative to one’s being-in-the-world and one’s being in language (where setsand concepts, such as the concept 'humanity', get their names). However, he continues, the dominations themselvesare, whilst being relative concepts, not necessarily intrinsic to language and constructible thought; rather one canaxiomatically define a domination – in the terms of mathematical ontology – as a set of conditions such that anycondition outside the domination is dominated by at least one term inside the domination. One does not necessarilyneed to refer to constructible language to conceive of a 'set of dominations’, which he refers to as the indiscernible set,or the generic set. It is therefore, he continues, possible to think beyond the strictures of the relativistic constructibleuniverse of language, by a process Cohen calls forcing. And he concludes in following that while ontology can markout a space for an inhabitant of the constructible situation to decide upon the indiscernible, it falls to the subject –about which the ontological situation cannot comment – to nominate this indiscernible, this generic point; and thusnominate, and give name to, the undecidable event. Badiou thereby marks out a philosophy by which to refute theapparent relativism or apoliticism in post-structuralist thought.Badiou’s ultimate ethical maxim is therefore one of: 'decide upon the undecidable'. It is to name the indiscernible, the

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generic set, and thus name the event that re-casts ontology in a new light. He identifies four domains in which a subject(who, it is important to note, becomes a subject through this process) can potentially witness an event: love, science,politics and art. By enacting fidelity to the event within these four domains one performs a 'generic procedure', whichin its undecidability is necessarily experimental, and one potentially recasts the situation in which being takes place.Through this maintenance of fidelity, truth has the potentiality to emerge.In line with his concept of the event, Badiou maintains, politics is not about politicians, but activism based on thepresent situation and the evental [sic] (his translators’ neologism) rupture. So too does love have this characteristic ofbecoming anew. Even in science the guesswork that marks the event is prominent. He vigorously rejects the tag of'decisionist' (the idea that once something is decided it 'becomes true'), but rather argues that the recasting of a truthcomes prior to its veracity or verifiability. As he says of Galileo (p. 401):

When Galileo announced the principle of inertia, he was still separated from the truth of the new physicsby all the chance encounters that are named in subjects such as Descartes or Newton. How could he, withthe names he fabricated and displaced (because they were at hand – 'movement', 'equal proportion', etc.),have supposed the veracity of his principle for the situation to-come that was the establishment of modernscience; that is, the supplementation of his situation with the indiscernible and unfinishable part that onehas to name 'rational physics’?

While Badiou is keen to reject an equivalence between politics and philosophy, he correlates nonetheless his politicalactivism and skepticism toward the parliamentary-democratic process with his philosophy, based around singular,situated truths, and potential revolutions.

1.4 L'Organisation Politique

Alain Badiou is a founding member (along with Natacha Michel and Sylvain Lazarus) of the militant French politicalorganisation L'Organisation Politique, which was active from 1985 until it disbanded in 2007.[19] It called itself apost-party organization concerned with direct popular intervention in a wide range of issues (including immigration,labor, and housing). In addition to numerous writings and interventions, L'Organisation Politique highlighted theimportance of developing political prescriptions concerning undocumented migrants (les sans papiers), stressing thatthey must be conceived primarily as workers and not immigrants.[20]

1.5 Works

1.5.1 Philosophy

• Le concept de modèle (1969, 2007)

• Théorie du sujet (1982)

• Peut-on penser la politique? (1985)

• L'Être et l'Événement (1988)

• Manifeste pour la philosophie (1989)

• Le nombre et les nombres (1990)

• D'un désastre obscur (1991)

• Conditions (1992)

• L'Éthique (1993)

• Deleuze (1997)

• Saint Paul. La fondation de l'universalisme (1997, 2002)

• Abrégé de métapolitique (1998)

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• Court traité d'ontologie transitoire (1998)

• Petit manuel d'inesthétique (1998)

• Le Siècle (2005)

• Logiques des mondes. L'être et l'événement, 2 (2006)

• Petit panthéon portatif (2008)

• Second manifeste pour la philosophie (2009)

• L'Antiphilosophie de Wittgenstein (2009)

• Éloge de l'Amour (2009)

• Heidegger. Le nazisme, les femmes, la philosophie co-authored with Barbara Cassin (2010)

• Il n'y a pas de rapport sexuel co-authored with Barbara Cassin (2010)

• La Philosophie et l'Événement interviews with Fabien Tarby (ed.) (2010)

• Cinq leçons sur le cas Wagner (2010)

• Le Fini et l'Infini (2010)

• La Relation énigmatique entre politique et philosophie (2011)

• La République de Platon (2012)

• L'aventure de la philosophie française (2012)

1.5.2 Critical essays

• L'autonomie du processus esthétique (1966)

• Rhapsodie pour le théâtre (1990)

• Beckett, l'increvable désir (1995)

• Cinéma (2010)

1.5.3 Literature and drama

• Almagestes (1964)

• Portulans (1967)

• L'Écharpe rouge (1979)

• Ahmed le subtil (1994)

• Ahmed Philosophe, followed by Ahmed se fâche (1995)

• Les Citrouilles, a comedy (1996)

• Calme bloc ici-bas (1997)

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1.5. WORKS 9

1.5.4 Political essays

• Théorie de la contradiction (1975)

• De l'idéologie with F. Balmès (1976)

• Le Noyau rationnel de la dialectique hégelienne with L. Mossot and J. Bellassen (1977)

• Circonstances 1: Kosovo, 11 Septembre, Chirac/Le Pen (2003)

• Circonstances 2: Irak, foulard, Allemagne/France (2004)

• Circonstances 3: Portées du mot " juif " (2005)

• Circonstances 4: De quoi Sarkozy est-il le nom ? (2007)

• Circonstances 5: L'hypothèse communiste (2009)

• Circonstances 6: Le Réveil de l'Histoire (2011)

• Circonstances 7: Sarkozy: pire que prévu, les autres : prévoir le pire (2012)

• Mao. De la pratique et de la contradiction with Slavoj Žižek (2008)

• Démocratie, dans quel état ? with Giorgio Agamben, Daniel Bensaïd, Wendy Brown, Jean-Luc Nancy, JacquesRancière, Kristin Ross and Slavoj Žižek (2009)

• L'Idée du communisme vol. 1 (London Conference, 2009) (Alain Badiou and Slavoj Žižek eds.), with JudithBalso, Bruno Bosteels, Susan Buck-Morss, Terry Eagleton, Peter Hallward, Michael Hardt, Minqi Li, Jean-LucNancy, Toni Negri, Jacques Rancière, Alessandro Russo, Roberto Toscano, Gianni Vattimo, Wang Hui andSlavoj Žižek (2010)

• L'Explication, conversation avec Aude Lancelin with Alain Finkielkraut (2010)

• L'Antisémitisme partout. Aujourd'hui en France with Eric Hazan (2011)

• L'Idée du communisme, vol. 2 (Berlin Conference, 2010), (Alain Badiou and Slavoj Žižek eds.) with GlynDaly, Saroj Giri, Gernot Kamecke, Janne Kurki, ArtemyMagun, KubaMajmurek, KubaMikurda, Toni Negri,Frank Ruda, Bülent Somay, Janek Sowa, G.M. Tamás, Henning Teschke, Jan Völker, CécileWinter and SlavojŽižek (2011)

1.5.5 Pamphlets and serial publications

• Contribution au problème de la construction d'un parti marxiste-léniniste de type nouveau, with Jancovici, Men-etrey, and Terray (Maspero 1970)

• Jean Paul Sartre (Éditions Potemkine 1980)

• Le Perroquet. Quinzomadaire d'opinion (1981–1990)

• La Distance Politique (1990–?)

1.5.6 English translations


• Manifesto for Philosophy, transl. by Norman Madarasz; (Albany: SUNY Press, 1999): ISBN 978-0-7914-4220-3 (paperback); ISBN 978-0-7914-4219-7 (hardcover)

• Deleuze: The Clamor of Being, transl. by Louise Burchill; (Minnesota University Press, 1999): ISBN 978-0-8166-3140-7 (paperback); ISBN 978-0-8166-3139-1 (library binding)

• Ethics: An Essay on the Understanding of Evil, transl. by Peter Hallward; (New York: Verso, 2000): ISBN978-1-85984-435-9 (paperback); ISBN 978-1-85984-297-3

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• On Beckett, transl. and ed. by Alberto Toscano and Nina Power; (London: Clinamen Press, 2003): ISBN978-1-903083-30-7 (paperback); ISBN 978-1-903083-26-0 (hardcover)

• Infinite Thought: Truth and the Return to Philosophy, transl. and ed. by Oliver Feltham & Justin Clemens;(London: Continuum, 2003): ISBN 978-0-8264-7929-7 (paperback); ISBN 978-0-8264-6724-9 (hardcover)

• Metapolitics, transl. by Jason Barker; (New York: Verso, 2005): ISBN 978-1-84467-567-8 (paperback); ISBN978-1-84467-035-2 (hardcover)

• Saint Paul: The Foundation of Universalism; transl. by Ray Brassier; (Stanford: Stanford University Press,2003): ISBN 978-0-8047-4471-3 (paperback); ISBN 978-0-8047-4470-6 (hardcover)

• Handbook of Inaesthetics, transl. by Alberto Toscano; (Stanford: Stanford University Press, 2004): ISBN978-0-8047-4409-6 (paperback); ISBN 978-0-8047-4408-9 (hardcover)

• Theoretical Writings, transl. by Ray Brassier; (New York: Continuum, 2004)[21]

• Briefings on Existence: A Short Treatise on Transitory Ontology, transl. by Norman Madarasz; (Albany: SUNYPress, 2005)

• Being and Event, transl. by Oliver Feltham; (New York: Continuum, 2005)

• Polemics, transl. by Steve Corcoran; (New York: Verso, 2007)

• The Century, transl. by Alberto Toscano; (New York: Polity Press, 2007)

• The Concept of Model: An Introduction to the Materialist Epistemology of Mathematics, transl. by Zachery LukeFraser & Tzuchien Tho; (Melbourne: re.press, 2007). Open Access[22]

• Number and Numbers (New York: Polity Press, 2008): ISBN 978-0-7456-3879-9 (paperback); ISBN 978-0-7456-3878-2 (hardcover)

• The Meaning of Sarkozy (New York: Verso, 2008): ISBN 978-1-84467-309-4 (hardcover) ISBN 978-1-84467-629-3 (paperback)

• Conditions, transl. by Steve Corcoran; (New York: Continuum, 2009): ISBN 978-0-8264-9827-4 (hardcover)

• Logics of Worlds: Being and Event, Volume 2, transl. by Alberto Toscano; (New York: Continuum, 2009):ISBN 978-0-8264-9470-2 (hardcover)

• Pocket Pantheon: Figures of Postwar Philosophy, transl. by David Macey; (New York: Verso, 2009): ISBN978-1-84467-357-5 (hardcover)

• Theory of the Subject, transl. by Bruno Bosteels; (New York: Continuum, 2009): ISBN 978-0-8264-9673-7(hardcover)

• Philosophy in the Present, (with Slavoj Žižek); (New York: Polity Press, 2010): ISBN 978-0-7456-4097-6(paperback)

• The Communist Hypothesis, transl. by David Macey and Steve Corcoran; (New York: Verso, 2010): ISBN978-1-84467-600-2 (hardcover)

• Five Lessons on Wagner, transl. by Susan Spitzer with an 'Afterword' by Slavoj Žižek; (New York: Verso,2010): ISBN 978-1-84467-481-7 (paperback)

• Second Manifesto for Philosophy, transl. by Louise Burchill (New York: Polity Press, 2011)

• Wittgenstein’s Antiphilosophy, transl. by Bruno Bosteels; (New York: Verso, 2011)

• The Rational Kernel of the Hegelian Dialectic, transl. by Tzuchien Tho; (Melbourne: re.press, 2011)

• The Rebirth of History: Times of Riots and Uprisings, transl. by Gregory Elliott; (New York: Verso, 2012):ISBN 978-1-84467-879-2

• In Praise of Love, (with Nicolas Truong); transl. by Peter Bush; (London: Serpent’s Tail, 2012)

• Philosophy for Militants, transl. by Bruno Bosteels; (New York: Verso, 2012)

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1.5. WORKS 11

• The Adventure of French Philosophy, transl. by Bruno Bosteels; (New York: Verso, 2012)

• Plato’s Republic : A Dialogue in 16 Chapters, transl. by Susan Spitzer; (New York : Columbia University Press,2013)

• The Incident at Antioch/L'Incident d'Antioche: A Tragedy in Three Acts / Tragédie en trois actes, transl. bySusan Spitzer; (New York : Columbia University Press, 2013)

• Badiou and the Philosophers : Interrogating 1960s French Philosophy, transl. and ed. by Tzuchien Tho andGiuseppe Bianco; (New York : Bloomsbury Academic, 2013)

• Philosophy and the Event, (with Fabian Tarby); transl. by Louise Burchill; (Malden, MA: Polity, 2013)

• Reflections on Anti-Semitism, (with Eric Hazan); transl. by David Fernbach; (London: Verso, 2013)

• Rhapsody for the Theatre, transl. and ed. by Bruno Bosteels; (London: Verso, 2013)

• Cinema, transl. by Susan Spitzer; (Malden, MA: Polity, 2013)

• Mathematics of the Transcendental: Onto-logy and being-there, transl. by A.J. Bartlett and Alex Ling; (London:Bloomsbury, 2014)

• Ahmed the Philosopher: Thirty-four Short Plays for Children and Everyone Else, transl. by Joseph Litvak; (NewYork : Columbia University Press, 2014)

• Jacques Lacan, Past and Present: A Dialogue, (with Elisabeth Roudinesco); transl. by Jason E. Smith; (NewYork: Columbia University Press, 2014)

• Controversies: Politics and Philosophy in our Time, (with Jean-Claude Milner); transl. by ?; (London: Polity,2014)

• Confrontation: A Conversation with Aude Lancelin, (with Alain Finkielkraut); transl. by Susan Spitzer; (Lon-don: Polity, 2014)

• The Age of the Poets: And Other Writings on Twentieth-Century Poetry and Prose, transl. by Bruno Bosteels;(New York: Verso, 2014)


• Badiou Studies

• “The Cultural Revolution: The Last Revolution?", transl. by Bruno Bosteels; positions: asia critique, Volume13, Issue 3, Winter 2005; (Durham: Duke University Press, 2005): ISSN 1067-9847

• “Selections from Théorie du sujet on the Cultural Revolution”, transl. by Alberto Toscano with the assistanceof Lorenzo Chiesa and Nina Power; positions: asia critique, Volume 13, Issue 3, Winter 2005; (Durham: DukeUniversity Press, 2005): ISSN 1067-9847

• “Further Selections from Théorie du sujet on the Cultural Revolution”, transl. by Lorenzo Chiesa; positions:asia critique, Volume 13, Issue 3, Winter 2005; (Durham: Duke University Press, 2005): ISSN 1067-9847

• “The Triumphant Restoration”, transl. by Alberto Toscano; positions: asia critique, Volume 13, Issue 3, Winter2005; (Durham: Duke University Press, 2005): ISSN 1067-9847

• “An Essential Philosophical Thesis: 'It Is Right to Rebel against the Reactionaries’", transl. by Alberto Toscano;positions: asia critique, Volume 13, Issue 3, Winter 2005; (Durham: Duke University Press, 2005): ISSN1067-9847

1.5.7 DVD

• Democracy and Disappointment: On the Politics of Resistance: Alain Badiou and Simon Critchley in Conver-sation, (Event Date: Thursday, 15 November 2007); Location: Slought Foundation, Conversations in TheorySeries | Organized by Aaron Levy | Studio: Microcinema in collaboration with Slought Foundation | DVDRelease Date: 26 August 2008

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1.6 Lectures

• “The Event as Creative Novelty”, European Graduate School. 2009

• “Interview with Alain Badiou” BBC HARDtalk. March 2009.

• Creative Thinking. Al-Quds University, Jerusalem, Palestine, 17 January 2009.

• “Is the Word Communism forever Doomed?". Miguel Abreu Gallery, New York, 6 November 2008.

• “Theatre et Philosophie.” with Martin Puchner & Bruno Bosteels. La Maison Française, New York University,New York, 7 November 2008.

• What is love? Sexuality and Desire, European Graduate School. 2008

• “Democracy and Disappointment: On the Politics of Resistance”, with Simon Critchley. Slought Foundation,Philadelphia, the Departments of Romance Languages, History, and English, and the Program in ComparativeLiterature at the University of Pennsylvania. 15 November 2007.

• “Destruction, Negation, Subtraction”, European Graduate School, August 2007.

• “Homage to Jacques Derrida”, University of California, Irvine, 1 March 2006 (RealPlayer).

• “Democracy, Politics, Theory and Philosophy”, European Graduate School, August 2006.

• “Ours is not a terrible situation.” with Simon Critchley. Labyrinth Books, New York, 6 March 2006.

• “Politics, Democracy and Philosophy: An Obscure Knot”, Walter Chapin Simpson Center for the Humanitiesat University of Washington 23 February 2006.

• “Political Perversion and Democracy”, European Graduate School, August 2004.

• “Panorama de la Filosofía Francesa Contemporánea” Biblioteca Nacional de Buenos Aires, 2004

• “Finkielkraut-Badiou: Le-Face-à-Vace” The Nouvel Obs (Transcript in French)[23]

• “Faut-il réinventer l'amour?" – Ce Soir. French television. En direct, France 3 (French)

1.7 Notes[1] Tzuchien Tho, Giuseppe Bianco, Badiou and the Philosophers: Interrogating 1960s French Philosophy, A&C Black, 2013,

pp. xvii.

[2] Tzuchien Tho, Giuseppe Bianco, Badiou and the Philosophers: Interrogating 1960s French Philosophy, A&C Black, 2013,pp. xviii–xix.

[3] François Regnault Homepage at Cahiers pour l'Analyse

[4] Biography at alain-badiou.jimdo.com

[5] Badiou Homepage at Concept and Form: The Cahiers pour l'Analyse and Contemporary French Thought

[6] Badiou, Alain (2010). “Part I: “WeAre Still the Contemporaries ofMay '68"". The Communist Hypothesis (pbk). translatedby David Macey and Steve Corcoran. Verso. p. 58. ISBN 978-1-84467-600-2.

[7] Badiou, Alain. “Jacques Lacan.” Pocket Pantheon. Trans. David Macey. London: Verso, 2009

[8] Badiou, Alain. “Louis Althusser.” Pocket Pantheon. Trans. David Macey. London: Verso, 2009

[9] “Quentin Meillassoux”. CIEFPC. Retrieved 24 January 2014.

[10] “Alain Badiou – Uses of the Word “Jew"". Lacan.com. Archived from the original on 25 May 2011. Retrieved 18 June2011.

[11] On that subject, see articles against Badiou by:

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1.7. NOTES 13

• Roger-Pol Droit (“LeMonde des livres”, 25 November 2005) and Frédéric Nef (“LeMonde des livres”, 23 December2005), and in defense of Badiou by: Daniel Bensaid (“Le Monde des Livres”, 26 January 2006);

against Badiou by:

• Claude Lanzmann, Jean-Claude Milner and Eric Marty (“Les Temps modernes”, Nov.-December 2005/January2006), and Meir Waintrater “L’Arche” February 2006: “Alain Badiou et les Juifs: Une violence insoutenable”, andthe answers by Alain Badiou and Cécile Winter followed by rejoinders by Claude Lanzmann and Eric Marty (“LesTemps modernes”, March–June 2006). See also Badiou’s response to Eric Marty

[12] Alain Badiou - Global Center for Advanced Studies

[13] Johannes Thumfart: Learning from Las Vegas: Badiou’s Platonism Today, in: The Symptom 9.

[14] See here Feltham and Clamens’s introduction in Badiou’s book Infinite Thought, Continuum (2004)

[15] See Badiou’s book Infinite Thought, Continuum (2004)

[16] See here Badiou’s comments in the introduction to the English version of Being and Event, Continuum (2005)

[17] Sokal, Alan and Jean Bricmont (1999) Fashionable Nonsense: Postmodern Intellectuals’ Abuse of Science Macmillan, ISBN9780312204075, p. 180

[18] Scruton, Roger (31 August 2012). “A Nothing Would do As Well”. Times Literary Supplement.

[19] See the organisation’s website at http://web.archive.org/web/20071028083920/http://www.orgapoli.net/

[20] “European Graduate School – Faculty Overview”. Egs.edu. Archived from the original on 24 June 2011. Retrieved 18June 2011.

[21] Includes:

• Mathematics and Philosophy: The Grand Style and the Little Style, (unpublished)• Philosophy and Mathematics: Infinity and the End of Romanticism, (from Conditions, Paris, Seuil, 1992).• The Question of Being Today, (from Briefings on Existence, )• Platonism and Mathematical Ontology, (from Briefings on Existence)• The Being of Number, (from Briefings on Existence)• One, Multiple, Multiplicities, (from Multitudes, 1, 2000)• Spinoza’s Closed Ontology, (from Briefings on Existence)• The Event as Trans-Being, (revised and expanded version of an essay of the same title from Briefings on Existence)• On Subtraction, (from Conditions, Paris, Seuil, 1992)• Truth: Forcing and the Unnameable, (from Conditions, Paris,Seuil, 1992)• Kant’s Subtractive Ontology, (from Briefings on Existence)• Eight Theses on the Universal, (from Jelica Sumic (ed.) Universal, Singulier, Subjet, Paris, Kimé, 2000)• Politics as a Truth Procedure, (from Metapolitics)• Being and Appearance, (from Briefings on Existence)• Notes Toward Thinking Appearance, (unpublished)• The Transcendental, (from a draft manuscript [now published] of Logiques des mondes, Paris, Seuil)• Hegel and the Whole, (from a draft manuscript [now published] of Logiques des mondes, Paris, Seuil)• Language, Thought, Poetry, (unpublished)

[22] Alain Badiou, The Concept of Model.

[23] The Nouvel Obs invited the philosophers Alain Finkielkraut and Alain Badiou, members of opposite political camps, to talkabout national identity. According to Aude Lancelin whomoderated the discussion, “it came to an ideological confrontationof rare violence”.

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1.8 Further reading

1.8.1 Secondary literature on Badiou’s work

in English (books)

• Jason Barker, Alain Badiou: A Critical Introduction, London, Pluto Press, 2002.

• Peter Hallward, Badiou: A Subject to Truth, Minneapolis, University of Minnesota Press, 2003.

• Peter Hallward (ed.), Think Again: Badiou and the Future of Philosophy”, London, Continuum, 2004.

• Paul Ashton (Editor), A. J. Bartlett (Editor), Justin Clemens (Editor): The Praxis of Alain Badiou; (Melbourne:re.press, 2006).

• Adam Miller, Badiou, Marion, and St. Paul: Immanent Grace, London, Continuum, 2008.

• Bruno Bosteels, Badiou and Politics, Durham, Duke University Press, 2011.

• Oliver Feltham, Alain Badiou: Live Theory, London, Continuum, 2008.

• Burhanuddin Baki, Badiou’s Being and Event and the Mathematics of Set Theory, London, Bloomsbury Aca-demic, 2015.

• Sam Gillespie, The Mathematics of Novelty: Badiou’s Minimalist Metaphysics, (Melbourne, Australia: re.press,2008) (details on re.press website) (Open Access)

• Adrian Johnston, Badiou, Žižek, and Political Transformations: The Cadence of Change, Evanston, Northwest-ern University Press, 2009, forthcoming.

• Gabriel Riera (Editor), Alain Badiou: Philosophy and its Conditions, Albany: New York, SUNY Press, 2005.

• Christopher Norris, Badiou’s Being and Event: A Reader’s Guide, London, Continuum, 2009.

• A.J. Bartlett & Justin Clemens (eds) " Badiou: Key Concepts,” London, Acumen, 2010.

• Alex Ling, Badiou and Cinema, Edinburgh, Edinburgh University Press, 2010.

• Ed Pluth, Badiou: A Philosophy of the New, Malden, Polity, 2010.

• A. J. Bartlett, Badiou and Plato: An education by truths, Edinburgh, Edinburgh University Press, 2011.

• P. M. Livingston, The Politics of Logic: Badiou, Wittgenstein, and the Consequences of Formalism, New York,Routledge, 2011.

• Steven Corcoran (ed.): The Badiou Dictionary, Edinburgh, Edinburgh University Press 2015, ISBN 978-0-7486-4096-6

In English (journals, essays and articles)

• Cantor, Lacan, Mao, Becket, meme combat: The philosophy of Alain Badiou essay by Jean-Jacques Lecercle

• Alain Badiou’s Theory of the Subject: Part 1. The Recommencement of Dialectical Materialism? by BrunoBosteels

• Society and Space Theme Issue: Being and Spatialization vol. 27. Issue 5. 2009, interview and articles by M.Constantinou, N. Madarasz, J. Flowers MacCannell (See: “Environment and Planning D: Society and Spacecontents vol 29”. Envplan.com. Archived from the original on 22 June 2011. Retrieved 18 June 2011.)

• Fatal Repetition: Badiou and the Age of the Poets, with Appendix, A Psychoanalysis of Alain Badiou, byJames Luchte, Istiraki (Turkey), 5 May 2014.

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In French (books)

• Charles Ramond (éd), Penser le multiple, Paris, Éditions L'Harmattan, 2002

• Fabien Tarby, La Philosophie d'Alain Badiou, Paris, Éditions L'Harmattan, 2005

• Fabien Tarby, Matérialismes d'aujourd'hui : de Deleuze à Badiou , Paris, Éditions L'Harmattan, 2005

• Eric Marty, Une Querelle avec Alain Badiou, philosophe, Paris, Editions Gallimard, coll. L'Infini, 2007

• Bruno Besana et Oliver Feltham (éd), Écrits autour de la pensée d'Alain Badiou, Paris, Éditions L'Harmattan,2007.

In Basque (books and articles)

• Antton Azkargorta (1996): “Hitzaurrea” in Alain Badiou, Etika, Bilbo, Besatari ISBN 84-921104-1-4

• Imanol Galfarsoro (2011): “Alain Badiou. Filosofia etiko-politikoa I”, hAUSnART, 0: 124–129

• Imanol Galfarsoro (2012): “Alain Badiou. Filosofia etiko-politikoa II”, hAUSnART, 1: 108–114

• Imanol Galfarsoro (2012): “Alain Badiou eta hipotesi komunistaren birdefinizioak”, hAUSnART, 2: 82–99

• Imanol Galfarsoro (2012): "(Post)Marxismoa, kultura eta eragiletasuna: Ibilbide historiko labur bat” in AlaitzAizpuru(koord.), Euskal Herriko pentsamenduaren gida, Bilbo, UEU. ISBN 978-84-8438-435-9

• Xabier Insausti & Irati Oliden (2012): Konpromisorik gabeko filosofia. Alain Badiou, Donostia, Jakin ISBN978-84-95234-44-5

• Alain Badiou on the Lapiko Kritikoa basque website.

In Spanish (books and articles)

• Carlos Gómez Camarena and Angelina Uzín Olleros (eds.), Badiou fuera de sus límites, Buenos Aires, ImagoMundi, 2010. ISBN 978-950-793-102-4

1.9 External links• Alain Badiou Faculty Page at European Graduate School

• Alain Badiou Bibliography at Lacan Dot Com

• Alain Badiou Archive at MidEastDilemma.com

• What is a philosophical Institution? or: Address, Transmission, Inscription. Cosmos and History: The Journalof Natural and Social Philosophy, Vol 2, No 1-2 (2006)

• LES REPONSES ECRITES D’ALAIN BADIOU Interviewed by Ata Hoodashtian, for Le journal PhilosophiePhilosophie, Université Paris VIII.

1.9.1 Critical opinions

• On Alain Badiou and Logiques des mondes by Slavoj Žižek

• The Marxist hypothesis: a response to Alain Badiou’s “communist hypothesis” by Chris Cutrone

• The Anarchist Hypothesis, or Badiou, Žižek, and the Anti-Anarchist Prejudice by Gabriel Kuhn

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Chapter 2

Andreas Blass

Andreas Raphael Blass (born October 27, 1947 in Nuremberg) is a mathematician, currently a professor at theUniversity of Michigan. He specializes in mathematical logic, particularly set theory, and theoretical computer sci-ence.Blass graduated from the University of Detroit, where he was a Putnam Fellow, in 1966 with a B.S. in physics.He received his Ph.D. in 1970 from Harvard University, with a thesis on Orderings of Ultrafilters written under thesupervision of Frank Wattenberg.[1] Since 1970 he has been employed by the University of Michigan, first as a T.H.Hildebrandt Research Instructor (1970–72), then assistant professor (1972–76), associate professor (1976–84) andsince 1984 he has been a full professor there.In 2014, he became a Fellow of the American Mathematical Society.[2]

2.1 Selected publications and results

In 1984 Blass proved that the existence of a basis for every vector space is equivalent to the Axiom of Choice. Hemade important contributions in the development of the set theory of the reals and forcing.Blass was the first to point out connections between game semantics and linear logic.He has authored about 175 research articles in mathematical logic and theoretical computer science, including:

• Blass, Andreas (1984). “Existence of bases implies the axiom of choice”. Axiomatic set theory, Proc. AMS-IMS-SIAM Jt. Summer Res. Conf., Boulder/Colo. 1983, Contemp. Math. 31. pp. 31–34.

• Blass, Andreas; Shelah, Saharon (1987). “There may be simple Pℵ1 - and Pℵ2 -points and the Rudin-Keislerordering may be downward directed”. Annals of Pure and Applied Logic 33: 213–243. doi:10.1016/0168-0072(87)90082-0.

• Blass, Andreas (1992). “A game semantics for linear logic”. Annals of Pure and Applied Logic 56: 183–220.doi:10.1016/0168-0072(92)90073-9.

• Blass, Andreas; Gurevich, Yuri (2003). “Algorithms: a quest for absolute definitions” (PDF). Bull. Eur. Assoc.Theor. Comput. Sci. EATCS 81: 195–225. Retrieved 2008-04-28.

2.2 References[1] Andreas Blass at the Mathematics Genealogy Project

[2] List of Fellows of the American Mathematical Society

2.3 External links• Blass’s page at UM


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Chapter 3

Bohuslav Balcar

Bohuslav Balcar (Czech pronunciation: [ˈboɦuslaf ˈbaltsar]; born 1943) is a Czech mathematician. He is a seniorresearcher at the Center for Theoretical Study (CTS), and a professor at Charles University in Prague. His researchinterests are mainly related to foundations of mathematics.Balcar received his Ph.D. in 1966 from Charles University.

3.1 External links• Balcar’s bio at CTS


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Chapter 4

Cesare Burali-Forti

Cesare Burali-Forti (13 August 1861 – 21 January 1931) was an Italian mathematician.He was born in Arezzo, and was an assistant of Giuseppe Peano in Turin from 1894 to 1896, during which time hediscovered what came to be called the Burali-Forti paradox of Cantorian set theory. He died in Turin.

4.1 Books by C. Burali-Forti• Analyse vectorielle générale: Applications à la mécanique et à la physique. with Roberto Marcolongo (Mattéi& co., Pavia, 1913).

• Corso di geometria analitico-proiettiva per gli allievi della R. Accademia Militare (G. B. Petrini di G. Gallizio,Torino, 1912).

• Geometria descrittiva (S. Lattes & c., Torino, 1921).

• Introduction à la géométrie différentielle, suivant la méthode de H. Grassmann (Gauthier-Villars,1897).

• Lezioni Di Geometria Metrico-Proiettiva (Fratelli Bocca, Torino, 1904).

• Meccanica razionale with Tommaso Boggio (S. Lattes & c.,Torino, 1921).

• Logica Matematica (Hoepli, Milano, 1894.

• Complete listing of publications and Bibliography, 8 pages.

4.2 Bibliography

Primary literature in English translation:

• Jean van Heijenoort, 1967. A Source Book in Mathematical Logic, 1879-1931. Harvard Univ. Press.

• 1897. “A question on transfinite numbers,” 104-11.• 1897. “On well-ordered classes,” 111-12.

Secondary literature:

• Ivor Grattan-Guinness, 2000. The Search for Mathematical Roots 1870-1940. Princeton Uni. Press.

4.3 References• O'Connor, John J.; Robertson, Edmund F., “Cesare Burali-Forti”, MacTutor History of Mathematics archive,University of St Andrews.


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4.4 External links• “Introduction to Differential Geometry, following the method of H. Grassmann” (English translation)

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Cesare Burali-Forti.

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Chapter 5

Heinz Bachmann

HeinzBachmann (born 1924) is amathematicianwhoworked at the Eidgenössische Sternwarte (federal observatory)in Zurich. He introduced the Bachmann-Howard ordinal and ordinal collapsing functions.

5.1 References• Bachmann, Heinz (1950), “Die Normalfunktionen und das Problem der ausgezeichneten Folgen von Ord-nungszahlen”, Vierteljschr. Naturforsch. Ges. Zürich 95: 115–147, MR 0036806

• Bachmann, Heinz (1955), Transfinite Zahlen, Ergebnisse der Mathematik und ihrer Grenzgebiete (N.F.), Heft1., Springer-Verlag, Berlin-Göttingen-Heidelberg, ISBN 978-3642885150, MR 0071481


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Chapter 6

James Earl Baumgartner

James Earl Baumgartner (March 23, 1943 – December 28, 2011) was an American mathematician who workedin set theory, mathematical logic and foundations, and topology.[1]

Baumgartner was born in Wichita, Kansas, began his undergraduate study at the California Institute of Technology in1960, then transferred to the University of California, Berkeley, from which he received his PhD in 1970 from for adissertation entitled Results and Independence Proofs in Combinatorial Set Theory. His advisor was Robert Vaught.[2]He became a professor at Dartmouth College in 1969, and there spent his entire career.One of Baumgartner’s results is the consistency of the statement that any two ℵ1 -dense sets of reals are order iso-morphic (a set of reals is ℵ1 -dense if it has exactly ℵ1 points in every open interval). With András Hajnal he provedthe result (Baumgartner–Hajnal theorem) that the partition relation ω1 → (α)2n holds for α < ω1, n < ω . He diedof a heart attack in 2011.[3]

6.1 See also

• Baumgartner’s axiom

6.2 Selected publications

• Baumgartner, James E., A new class of order types, Annals of Mathematical Logic, 9:187–222, 1976

• Baumgartner, James E., Ineffability properties of cardinals I, Infinite and Finite Sets, Keszthely (Hungary) 1973,volume 10 of Colloquia Mathematica Societatis János Bolyai, pages 109–130. North-Holland, 1975

• Baumgartner, James E.; Harrington, Leo; Kleinberg, Eugene, Adding a closed unbounded set, Journal of Sym-bolic Logic, 41(2):481–482, 1976

• Baumgartner, James E., Ineffability properties of cardinals II, Robert E. Butts and Jaakko Hintikka, editors,Logic, Foundations of Mathematics and Computability Theory, pages 87–106. Reidel, 1977

• Baumgartner, James E.; Galvin, Fred, Generalized Erdős cardinals and 0#, Annals of Mathematical Logic 15,289–313, 1978

• Baumgartner, James E.; Erdős, Paul; Galvin, Fred; Larson, J., Colorful partitions of cardinal numbers, Can. J.Math. 31, 524–541, 1979

• Baumgartner, James E.; Erdős, Paul; Higgs, D., Cross-cuts in the power set of an infinite set, Order 1, 139–145,1984

• Baumgartner, James E. (Editor), Axiomatic Set Theory (Contemporary Mathematics, Volume 31), 1990


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6.3 References[1] “James E. Baumgartner Obituary”. Rand-wilson.com. Retrieved 2012-01-06.

[2] James Earl Baumgartner at the Mathematics Genealogy Project


• Valley News obituary

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Chapter 7

John Lane Bell

John Lane Bell (born March 25, 1945) is Professor of Philosophy at the University of Western Ontario in Canada.He has made contributions to mathematical logic and philosophy, and is the author of a number of books. Hisresearch includes such topics as set theory, model theory, lattice theory, modal logic, quantum logic, constructivemathematics, type theory, topos theory, infinitesimal analysis, spacetime theory, and the philosophy of mathematics.He is the author of more than 70 articles and of 11 books. In 2009, he was elected a Fellow of the Royal Society ofCanada.He was awarded a scholarship to Oxford University at the age of 15, and graduated with a D.Phil. in Mathemat-ics: his dissertation supervisor was John Crossley. During 1968-89 he was Lecturer in Mathematics and Reader inMathematical Logic at the London School of Economics.[1]

John Bell’s students include Graham Priest (Ph.D. Mathematics LSE, 1972), Michael Hallett (Ph.D. Philosophy LSE,1979), David DeVidi (Ph.D. Philosophy UWO, 1994), Elaine Landry (Ph.D. Philosophy UWO, 1997) and RichardFeist (Ph.D. Philosophy UWO, 1999).

7.1 Bibliography

7.1.1 Books

• Intuitionistic Set Theory. College Publications, 2013.

• Set Theory: Boolean-Valued Models and Independence Proofs. Oxford University Press 2011.

• The Axiom of Choice. College Publications, 2009.

• The Continuous and the Infinitesimal in Mathematics and Philosophy. Polimetrica, 2005.

• (With D. DeVidi and G. Solomon) Logical Options: An Introduction to Classical and Alternative Logics.Broadview Press, 2001.

• The Art of the Intelligible: An Elementary Survey of Mathematics in its Conceptual Development. Kluwer, 1999.

• A Primer of Infinitesimal Analysis. Cambridge University Press, 1998. Second Edition, 2008.

• Toposes & Local Set Theories: An Introduction. Clarendon Press, Oxford, 1988. Reprinted by Dover, 2008.

• Boolean-Valued Models and Independence Proofs in Set Theory. Clarendon Press, Oxford, 1977. 2nd edition,1985. 3rd edition, 2005.

• (With M. Machover). A Course in Mathematical Logic. North-Holland, Amsterdam, 1977. 4th printing, 2003.

• (WithA. B. Slomson). Models andUltraproducts: An Introduction. North-Holland, Amsterdam, 1969. Reprintedby Dover, 2006.


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7.1.2 Journal articles

• The Axiom of Choice in the Foundations of Mathematics, forthcoming in volume on Foundations of Mathe-matics, Giovanni Sommaruga, ed., University of Western Ontario Series, Springer

• Cohesiveness, Intellectica, 41, 2009.

• Types, Sets and Categories, Handbook of the History of Logic (Elsevier), forthcoming.

• Hermann Weyl, Stanford Encyclopedia of Philosophy, 2009

• “The Axiom of Choice and the Law of ExcludedMiddle inWeak Set Theories”,Mathematical Logic Quarterly,54, no. 2, 2008.

• “The Axiom of Choice”, Stanford Encyclopedia of Philosophy, 2008.

• Contribution to “Philosophy of Mathematics: 5 Questions”, V. Hendricks and H. Leitgeb, eds., AutomaticPress, 2007.

• “Incompleteness in a General Setting”. Bulletin of Symbolic Logic 13, 2007.

• “Cover Schemes, Frame-Valued Sets and Their Potential Uses in Spacetime Physics”. Spacetime Physics Re-search Trends, Horizons in World Physics, Volume 248, Nova Science Publishers, New York, 2007.

• “Cosmological Theories and theQuestion of the Existence of a Creator”. Religion and the Challenges of Science,Ashgate Publishers, 2007.

• “Abstract and Variable Sets in Category Theory”. In What is Category Theory?, Polimetrica, 2006.

• “Divergent Concepts of the Continuum in 19th and Early 20th Century Mathematics and Philosophy”. Ax-iomathes 15, 2005.

• “The Development of Categorical Logic”, Handbook of Philosophical Logic, Volume 12. Springer, 2005.

• “Continuity and Infinitesimals”. Stanford Encyclopedia of Philosophy, 2005.

• “Choice Principles in Intuitionistic Set Theory.”, A Logical Approach to Philosophy, Essays in Honour of Gra-ham Solomon, D. DeVidi and T. Kenyon, eds., Springer, 2006.

• “Oppositions and Paradoxes in Mathematics and Philosophy.” Axiomathes 15, 2005.

• “Observations onMathematics”,Mathematics as Story, Proceedings of 2003 Fields Institute Conference, UWO,2004.

• Whole and Part in Mathematics. Axiomathes 14, 2004.

• (With Geoffrey Hellman) “Pluralism and the Foundations of Mathematics”. Proceedings of Workshop on Sci-entific Pluralism, University of Minnesota, 2002. Minnesota University Press, 2006.

• “SomeNew Intuitionistic Equivalents of Zorn’s Lemma”,Archive forMathematical Logic, 42, Number 8, 2003.

• “Russell’s Paradox and Diagonalization in a Constructive Context”, 100 Years of Russell’s Paradox, Munich2001, Walter de Gruyter, 2004.

• “HermannWeyl’s Later Philosophical Views: His Divergence fromHusserl”,Husserl and the Sciences, R. Feist,ed. U. of Ottawa Press, 2003.

• “The Development of Categorical Logic”, Handbook of Philosophical Logic, Volume 12. Springer, 2005.

• “Time and Causation in Gödel’s Universe”, Transcendent Philosophy 3, 2002.

• “Observations on Category Theory”, Axiomathes 12, 2001

• “The Continuum in Smooth Infinitesimal Analysis”. In Reuniting the Antipodes-Constructive and NonstandardViews of the Continuum. Symposion Proceedings, San Servolo/Venice, Italy, 1999. U. Berger, H. Osswald andP. Schuster, eds. Kluwer, 2001.

• “Continuity and the Logic of Perception”, Transcendent Philosophy 1, no. 2, 2000.

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• “Hermann Weyl on Intuition and the Continuum”, Philosophia Mathematica (3), 8, 2000.

• “Sets and Classes as Many”, Journal of Philosophical Logic, 29, 2000.

• “Infinitary Logic”, Stanford Encyclopedia of Philosophy, 2000

• “Finite Sets and Frege Structures”, Journal of Symbolic Logic, 64, no. 4,1999.

• “Frege’s Theorem in a Constructive Setting”, Journal of Symbolic Logic, 64, no. 2, 1999.

• “Boolean Algebras and Distributive Lattices Treated Constructively”, Math. Logic Quarterly 45, 1999.

• “Boolean Algebras”, Routledge Encyclopedia of Philosophy, 1998.

• “Zorn’s Lemma and Complete Boolean Algebras in Intuitionistic Type Theories”, Journal of Symbolic Logic62, no. 4, 1997.

• (With S. Gebellato) “Precovers, Modalities, and Universal Closure Operators in a Topos”, Math. Logic Quar-terly 42, 1996.

• “Polymodal Lattices and Polymodal Logic”, Math. Logic Quarterly 42, 1996.

• (With W. Demopoulos) “Elementary Propositions and Independence”, Notre Dame J. of Formal Logic, 37, no.1, 1996.

• “Logical Reflections on the Kochen-Specker Theorem”, in Perspectives on Quantum Reality, R. Clifton, ed.,Kluwer, 1996.

• (With R.Clifton†) “Quasi Boolean Algebras and Simultaneously Definite Properties in Quantum Mechanics”,Int. Journal of Theoretical Physics, 34, 12, 1995.

• “Infinitesimals and the Continuum”, Mathematical Intelligencer, 17, no. 2, 1995.

• “Type-Reducing Correspondences and Well-Orderings: Frege’s and Zermelo’s Constructions Re-examined”,Journal of Symbolic Logic, 60, no. 1, 1995.

• “Frege’s Theorem and the Zermelo-Bourbaki Lemma”. Appendix to Frege’s Philosophy of Mathematics, W.Demopoulos, ed. Harvard U.P., 1995

• “Fregean Extensions of First-Order Theories”, Math. Logic Quarterly, 40, 1994. (Also reprinted in W. De-mopoulos, ed. Frege’s Philosophy of Mathematics, Harvard U.P. 1995)

• “Hilbert’s Epsilon Operator in Intuitionistic Type Theories”, Math. Logic Quarterly, 39, 1993.

• (with W. Demopoulos) “Frege’s Theory of Concepts and Objects and the Interpretation of Second-OrderLogic”, Philosophia Mathematica, (3), 1, 1993.

• “Hilbert’s Epsilon-Operator and Classical Logic”, Journal of Philosophical Logic, 22, 1993.

• “Some Propositions Equivalent to the Sikorski Extension Theorem for Boolean Algebras”, Fundamenta Math-ematicae 130 (1988).

• “Infinitesimals”, Synthese, 75, 1988.

• “Logic, the Paradoxes, and the Foundations of Mathematics”, LSE Quarterly Vol.I, No.3, 1987.

• “From Absolute to Local Mathematics”, Synthese 69, 1986.

• “A New Approach to Quantum Logic”, Brit. J. Phil. Sc., 37, 1986.

• “Orthospaces and Quantum Logic”, Foundations of Physics 15, 1985.

• “Orthologic, Forcing and the Manifestation of Attributes”, Proceedings of 1981 S.E. Asian Conference in Math-ematical Logic. North Holland, Amsterdam, 1983.

• “The Strength of the Sikorski Extension Theorem for Boolean Algebras”, Journal of Symbolic Logic 48, 1983.

• (With M.F. Hallett), “Logic, Quantum Logic, and Empiricism”, Philosophy of Science 49, 1982.

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• “Categories, Toposes and Sets”, Synthese, 51, No.3, 1982.

• “Some Aspects of the Category of Subobjects of Constant Objects in a Topos”, Journal of Pure and AppliedAlgebra 24, 1982.

• “Category Theory and the Foundations of Mathematics”, Brit.J.Phil.Sci. 32, 1981.

• “Isomorphism of Structures in S-Toposes”, Journal of Symbolic Logic, 46, 1981.

• “The Infinite Past Regained: A Reply to Whitrow”, Brit.J.Phil.Sci. Sci, 1979

• “Boolean Extensions as Toposes”, Bull. de la Soc. Francaise de Logique, Methodologie et Phil.des Sci. 6,1979.

• “Uncountable Standard Models of ZFC + V = L”, in Set Theory and Hierarchy Theory, a Memorial Tribute toAndrzej Mostowski, Springer Lecture Notes in Math. 537,1976.

• “A Note on Generic Ultrafilters”, Zeitschr. f. Math.Logik und Grund.der Math. 22, 1976.

• “Universal Complete Boolean Algebras and Cardinal Collapsing”, Zeitsch. f. Math.Logik und Grund. derMath. 22, 1976.

• “A Characterization of Universal Complete Boolean Algebras”, J. London Math.Soc. (2), 12, 1975.

• “On Compact Cardinals”, Zeitschr.f. Math.Logik und Grund.der Math. 20.1974.

• (With D.H. Fremlin), “A Geometric Form of the Axiom of Choice”, Fund. Math. LXXVII, 1972.

• (With D.H. Fremlin), “The Maximal Ideal Theorem for Lattices of Sets”, Bull. London Math. Soc., 4, 1972.

• “On the Relationship between Weak Compactness and Restricted Second- Order Languages”, Arch. Math.Logik 15, 1972.

• “Some Remarks on Current Mathematical Practice”, in Proceedings of the Bertrand Russell Memorial LogicConference, Denmark, 1971.

• (With F. Jellett). “On the Relationship between the Boolean Prime Ideal Theorem and Two Principles ofFunctional Analysis”, Bull. de l'Acad. Pol. des Sci., XIX, No.3, 1971.

• “Weak Compactness in Restricted Second-Order Languages”, Bull. de l'Acad. Pol. des Sci., No.3, 1970.

7.2 References[1] “Professor John L. Bell”. University of Western Ontario. Retrieved 25 March 2010.

7.3 External links• John Bell’s webpage

• John Bell at the Mathematics Genealogy Project

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Chapter 8

L. E. J. Brouwer

Luitzen Egbertus Jan Brouwer ForMemRS[3] (Dutch: [ˈlœyt̯sə(n) ɛɣˈbɛrtəs jɑn ˈbrʌu̯ər]; 27 February 1881 – 2December 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematicianand philosopher, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theoryand complex analysis.[1][4][5] He was the founder of the mathematical philosophy of intuitionism.

8.1 Biography

Early in his career, Brouwer proved a number of theorems that were breakthroughs in the emerging field of topology.The most celebrated result was his proof of the topological invariance of dimension. Among his further results, theBrouwer fixed point theorem is also well known. Brouwer also proved the simplicial approximation theorem in thefoundations of algebraic topology, which justifies the reduction to combinatorial terms, after sufficient subdivision ofsimplicial complexes, of the treatment of general continuous mappings. In 1912, at age 31, he was elected a memberof the Royal Netherlands Academy of Arts and Sciences.[6]

Brouwer in effect founded themathematical philosophy of intuitionism as an opponent to the then-prevailing formalismof David Hilbert and his collaborators Paul Bernays, Wilhelm Ackermann, John von Neumann and others (cf.Kleene (1952), p. 46–59). As a variety of constructive mathematics, intuitionism is essentially a philosophy ofthe foundations of mathematics.[7] It is sometimes and rather simplistically characterized by saying that its adherentsrefuse to use the law of excluded middle in mathematical reasoning.Brouwer was a member of the Significs group. It formed part of the early history of semiotics—the study of symbols—around Victoria, Lady Welby in particular. The original meaning of his intuitionism probably can not be completelydisentangled from the intellectual milieu of that group.In 1905, at the age of 24, Brouwer expressed his philosophy of life in a short tract Life, Art andMysticism described byDavis as “drenched in romantic pessimism” (Davis (2002), p. 94). Arthur Schopenhauer had a formative influence onBrouwer, not least because he insisted that all concepts be fundamentally based on sense intuitions.[8][9][10] Brouwerthen “embarked on a self-righteous campaign to reconstruct mathematical practice from the ground up so as to satisfyhis philosophical convictions"; indeed his thesis advisor refused to accept his Chapter II " 'as it stands, ... all interwovenwith some kind of pessimism and mystical attitude to life which is not mathematics, nor has anything to do with thefoundations of mathematics’ " (Davis, p. 94 quoting van Stigt, p. 41). Nevertheless, in 1908:

"... Brouwer, in a paper entitled “The untrustworthiness of the principles of logic”, challenged the beliefthat the rules of the classical logic, which have come down to us essentially from Aristotle (384-−322B.C.) have an absolute validity, independent of the subject matter to which they are applied” (Kleene(1952), p. 46).

“After completing his dissertation (1907 - see Van Dalen), Brouwer made a conscious decision to temporarily keephis contentious ideas under wraps and to concentrate on demonstrating his mathematical prowess” (Davis (2000), p.95); by 1910 he had published a number of important papers, in particular the Fixed Point Theorem. Hilbert—theformalist with whom the intuitionist Brouwer would ultimately spend years in conflict—admired the young man andhelped him receive a regular academic appointment (1912) at the University of Amsterdam (Davis, p. 96). It wasthen that “Brouwer felt free to return to his revolutionary project which he was now calling intuitionism " (ibid).


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He was combative for a young man. He was involved in a very public and eventually demeaning controversy in thelater 1920s with Hilbert over editorial policy at Mathematische Annalen, at that time a leading learned journal. Hebecame relatively isolated; the development of intuitionism at its source was taken up by his student Arend Heyting.Dutch mathematician and historian of mathematics, Bartel Leendert van der Waerden attended lectures given byBrouwer in later years, and commented: “Even though his most important research contributions were in topology,Brouwer never gave courses in topology, but always on—and only on—the foundations of his intuitionism. It seemedthat he was no longer convinced of his results in topology because they were not correct from the point of view ofintuitionism, and he judged everything he had done before, his greatest output, false according to his philosophy.”[11]

About his last years, Davis (2002) remarks:

"...he felt more and more isolated, and spent his last years under the spell of 'totally unfounded financialworries and a paranoid fear of bankruptcy, persecution and illness.' He was killed in 1966 at the age of85, struck by a vehicle while crossing the street in front of his house.” (Davis, p. 100 quoting van Stigt.p. 110.)

8.2 Bibliography

8.2.1 Primary literature in English translation

• Jean van Heijenoort, 1967 3rd printing 1976 with corrections, A Source Book in Mathematical Logic, 1879-1931. Harvard University Press, Cambridge MA, ISBN 0-674-32449-8 pbk. The original papers are prefacedwith valuable commentary.

• 1923. L. E. J. Brouwer: “On the significance of the principle of excluded middle in mathematics, espe-cially in function theory.” With two Addenda and corrigenda, 334-45. Brouwer gives brief synopsis ofhis belief that the law of excluded middle cannot be “applied without reservation even in the mathematicsof infinite systems” and gives two examples of failures to illustrate his assertion.

• 1925. A. N. Kolmogorov: “On the principle of excluded middle”, pp. 414–437. Kolmogorov supportsmost of Brouwer’s results but disputes a few; he discusses the ramifications of intuitionism with respectto “transfinite judgements”, e.g. transfinite induction.

• 1927. L. E. J. Brouwer: “On the domains of definition of functions”. Brouwer’s intuitionistic treatmentof the continuum, with an extended commentary.

• 1927. David Hilbert: “The foundations of mathematics,” 464-80• 1927. L. E. J. Brouwer: “Intuitionistic reflections on formalism,” 490-92. Brouwer lists four topics onwhich intuitionism and formalism might “enter into a dialogue.” Three of the topics involve the law ofexcluded middle.

• 1927. HermannWeyl: “Comments on Hilbert’s second lecture on the foundations of mathematics,” 480-484. In 1920 Weyl, Hilbert’s prize pupil, sided with Brouwer against Hilbert. But in this address Weyl“while defending Brouwer against some of Hilbert’s criticisms...attempts to bring out the significance ofHilbert’s approach to the problems of the foundations of mathematics.”

• Ewald, William B., ed., 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols.Oxford Univ. Press.

• 1928. “Mathematics, science, and language,” 1170-85.• 1928. “The structure of the continuum,” 1186-96.• 1952. “Historical background, principles, and methods of intuitionism,” 1197-1207.

• Brouwer, L. E. J., Collected Works, Vol. I, Amsterdam: North-Holland, 1975.

• Brouwer, L. E. J., Collected Works, Vol. II, Amsterdam: North-Holland, 1976.

• Brouwer, L. E. J., “Life, Art, and Mysticism,” Notre Dame Journal of Formal Logic, vol. 37 (1996), pp. 389–429. Translated by W. P. van Stigt with an introduction by the translator, pp. 381–87. Davis quotes from thiswork, “a short book... drenched in romantic pessimism” (p. 94).

• W. P. van Stigt, 1990, Brouwer’s Intuitionism, Amsterdam: North-Holland, 1990

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8.2.2 Secondary

• Dirk van Dalen, Mystic, Geometer, and Intuitionist: The Life of L. E. J. Brouwer. Oxford Univ. Press.

• 1999. Volume 1: The Dawning Revolution.• 2005. Volume 2: Hope and Disillusion.• 2013. L. E. J. Brouwer: Topologist, Intuitionist, Philosopher. HowMathematics is Rooted in Life. London:Springer (based on previous work).

• Martin Davis, 2000. The Engines of Logic, W. W. Norton, London, ISBN 0-393-32229-7 pbk. Cf. ChapterFive: “Hilbert to the Rescue” wherein Davis discusses Brouwer and his relationship with Hilbert andWeyl withbrief biographical information of Brouwer. Davis’s references include:

• Stephen Kleene, 1952 with corrections 1971, 10th reprint 1991, Introduction to Metamathematics, North-Holland Publishing Company, Amsterdam Netherlands, ISBN 0-7204-2103-9. Cf. in particular Chapter III:A Critique of Mathematical Reasoning, §13 “Intuitionism” and §14 “Formalism”.

• Koetsier, Teun, Editor, Mathematics and the Divine: A Historical Study, Amsterdam: Elsevier Science andTechnology, 2004, ISBN 0-444-50328-5

8.3 See also

• Gerrit Mannoury

• George F C Griss

8.4 References[1] L. E. J. Brouwer at the Mathematics Genealogy Project

[2] van Atten, Mark, “Luitzen Egbertus Jan Brouwer”, The Stanford Encyclopedia of Philosophy (Spring 2012 Edition).

[3] Kreisel, G.; Newman, M. H. A. (1969). “Luitzen Egbertus Jan Brouwer 1881–1966”. Biographical Memoirs of Fellows ofthe Royal Society 15: 39. doi:10.1098/rsbm.1969.0002.

[4] O'Connor, John J.; Robertson, Edmund F., “L. E. J. Brouwer”,MacTutor History of Mathematics archive, University of StAndrews.

[5] Luitzen Egbertus Jan Brouwer entry by Mark van Atten in the Stanford Encyclopedia of Philosophy

[6] “Luitzen E.J. Brouwer (1881 - 1966)". Royal Netherlands Academy of Arts and Sciences. Retrieved 21 July 2015.

[7] L. E. J. Brouwer (trans. by Arnold Dresden) (1913). “Intuitionism and Formalism”. Bull. Amer. Math. Soc. 20 (2):81–96. doi:10.1090/s0002-9904-1913-02440-6. MR 1559427.

[8] "...Brouwer and Schopenhauer are in many respects two of a kind.” Teun Koetsier, Mathematics and the Divine, Chapter30, “Arthur Schopenhauer and L.E.J. Brouwer: A Comparison,” p. 584.

[9] Brouwer wrote that “the original interpretation of the continuum of Kant and Schopenhauer as pure a priori intuition canin essence be upheld.” (Quoted in Vladimir Tasić's Mathematics and the roots of postmodernist thought, § 4.1, p. 36)

[10] “Brouwer’s debt to Schopenhauer is fully manifest. For both, Will is prior to Intellect.” [see T. Koetsier. “Arthur Schopen-hauer and L.E.J. Brouwer, a comparison,” Combined Proceedings for the Sixth and Seventh Midwest History of Mathe-matics Conferences, pages 272–290. Department of Mathematics, University of Wisconsin-La Crosse, La Crosse, 1998.].(Mark van Atten and Robert Tragesser, “Mysticism and mathematics: Brouwer, Gödel, and the common core thesis,”Published in W. Deppert and M. Rahnfeld (eds.), Klarheit in Religionsdingen, Leipzig: Leipziger Universitätsverlag 2003,pp.145–160)

[11] “Interview with B L van der Waerden, reprinted in AMSMarch 1997” (PDF). American Mathematical Society. Retrieved13 November 2015.

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8.5 External links• Life, Art and Mysticism written by L.E.J. Brouwer

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Chapter 9

Paul Bernays

Paul Isaac Bernays (17 October 1888 – 18 September 1977) was a Swiss mathematician, who made significantcontributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistantand close collaborator of David Hilbert.

9.1 Biography

Bernays spent his childhood in Berlin, and attended the Köllner Gymnasium, 1895-1907. At the University of Berlin,he studied mathematics under Issai Schur, Edmund Landau, Ferdinand Georg Frobenius, and Friedrich Schottky;philosophy under Alois Riehl, Carl Stumpf and Ernst Cassirer; and physics under Max Planck. At the University ofGöttingen, he studied mathematics under David Hilbert, Edmund Landau, Hermann Weyl, and Felix Klein; physicsunder Voigt and Max Born; and philosophy under Leonard Nelson.In 1912, the University of Berlin awarded him a Ph.D. in mathematics, for a thesis, supervised by Landau, on the ana-lytic number theory of binary quadratic forms. That same year, the University of Zurich awarded him the Habilitationfor a thesis on complex analysis and Picard’s theorem. The examiner was Ernst Zermelo. Bernays was Privatdozentat the University of Zurich, 1912–17, where he came to know George Pólya.Starting in 1917, DavidHilbert employed Bernays to assist himwith his investigations of the foundations of arithmetic.Bernays also lectured on other areas of mathematics at the University of Göttingen. In 1918, that university awardedhim a second Habilitation, for a thesis on the axiomatics of the propositional calculus of Principia Mathematica.[1]

In 1922, Göttingen appointed Bernays extraordinary professor without tenure. His most successful student there wasGerhard Gentzen. In 1933, he was dismissed from this post because of his Jewish ancestry. After working privatelyfor Hilbert for six months, Bernays and his family moved to Switzerland, whose nationality he had inherited fromhis father, and where the ETH employed him on occasion. He also visited the University of Pennsylvania and was avisiting scholar at the Institute for Advanced Study in 1935-36 and again in 1959-60.[2]

9.2 Mathematical work

Bernays’s collaboration with Hilbert culminated in the two volume work Grundlagen der Mathematik by Hilbert andBernays (1934, 1939), discussed in Sieg and Ravaglia (2005). In seven papers, published between 1937 and 1954 inthe Journal of Symbolic Logic, republished in (Müller 1976), Bernays set out an axiomatic set theory whose startingpoint was a related theory John von Neumann had set out in the 1920s. Von Neumann’s theory took the notion offunction as primitive; Bernays recast Von Neumann’s theory so that sets and proper classes were primitive. Bernays’stheory, with some modifications by Kurt Gödel, is now known as the Von Neumann–Bernays–Gödel set theory. Aproof from theGrundlagen der Mathematik that a sufficiently strong consistent theory cannot contain its own referencefunctor is now known as the Hilbert–Bernays paradox.


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9.3 Publications• Hilbert, David; Bernays, Paul (1934), Grundlagen der Mathematik. I, Die Grundlehren der mathematischenWissenschaften 40, Berlin, New York: Springer-Verlag, ISBN 978-3-540-04134-4, JFM 60.0017.02, MR0237246[3]

• Hilbert, David; Bernays, Paul (1939), Grundlagen der Mathematik. II, Die Grundlehren der mathematischenWissenschaften 50, Berlin, New York: Springer-Verlag, ISBN 978-3-540-05110-7, JFM 65.0021.02, MR0272596

• Bernays, Paul (1958), Axiomatic set theory, Studies in Logic and the Foundations of Mathematics, Amsterdam:North-Holland, ISBN 978-0-486-66637-2, MR 0106178

• Bernays, Paul (1976),Abhandlungen zur Philosophie derMathematik (inGerman), Darmstadt: WissenschaftlicheBuchgesellschaft, ISBN 978-3-534-06706-0, MR 0444417

9.4 Notes[1] Zach, Richard (1999). “Completeness before Post: Bernays, Hilbert, and the development of propositional logic”. Bulletin

of Symbolic Logic 5: 331–366. doi:10.2307/421184. Retrieved 26 November 2014.

[2] Institute for Advanced Study: A Community of Scholars

[3] MacLane, Saunders (1935). “Review: Grundlagen der Mathematik, Volume I. By D. Hilbert and P. Bernays” (PDF). Bull.Amer. Math. Soc. 41 (3): 162–165. doi:10.1090/s0002-9904-1935-06048-3.

9.5 References• Kneebone, Geoffrey, 1963. Mathematical Logic and the Foundation of Mathematics. Van Nostrand. Doverreprint, 2001. A gentle introduction to some of the ideas in the Grundlagen der Mathematic.

• Müller, Gert H., ed. (1976), Sets and classes. On the work by Paul Bernays, Studies in Logic and the Founda-tions of Mathematics 84, Amsterdam: North-Holland, ISBN 978-0-444-10907-1, MR 0414355

• Lauener, Henri (1978), “Paul Bernays (1888-−1977)", Zeitschrift für Allgemeine Wissenschaftstheorie 9 (1):13–20, doi:10.1007/BF01801939, ISSN 0044-2216, MR 546580

• Sieg, Wilfried; Ravaglia, Mark (2005), “Chapter 77. David Hilbert and Paul Bernays, Grundlagen der Math-ematik”, in Grattan-Guinness, Ivor, Landmark writings in western mathematics 1640-−1940, Elsevier B. V.,Amsterdam, pp. 981–99, doi:10.1016/B978-044450871-3/50158-3, ISBN978-0-444-50871-3, MR2169816

• Bernays and Set Theory, Akihiro Kanamori, The Bulletin of Symbolic Logic, Vol. 15, No. 1 (Mar., 2009),pp. 43–69.

9.6 External links• Hilbert Bernays Project

• O'Connor, John J.; Robertson, Edmund F., “Paul Bernays”,MacTutor History ofMathematics archive, Universityof St Andrews.

• Paul Bernays, Paul Bernays: A Short Biography (1976)

• Paul Bernays at the Mathematics Genealogy Project

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Chapter 10

Peter Aczel

Peter Henry George Aczel is a British mathematician, logician and Emeritus joint Professor in the School of Com-puter Science and the School of Mathematics at the University of Manchester.[7] He is known for his work in non-well-founded set theory,[8] constructive set theory,[9][10] and Frege structures.[11][12][13]

10.1 Education

Aczel completed his Bachelor of Arts in Mathematics in 1963[14] followed by a DPhil at the University of Oxford in1966 under the supervision of John Crossley.[7][15]

10.2 Career and research

After two years of visiting positions at the University of Wisconsin–Madison and Rutgers University Aczel took aposition at the University of Manchester. He has also held visiting positions at the University of Oslo, CaliforniaInstitute of Technology, Utrecht University, Stanford University and Indiana University Bloomington.[14] He was avisiting scholar at the Institute for Advanced Study in 2012.[16]

Aczel is on the editorial board of the Notre Dame Journal of Formal Logic[17] and the Cambridge Tracts in TheoreticalComputer Science, having previously served on the editorial boards of the Journal of Symbolic Logic and the Annalsof Pure and Applied Logic.[14][18]

10.3 References[1] Belo, Joao Filipe Castel-Branco (2008). Foundations of dependently sorted logic (PhD thesis). University of Manchester.

[2] Fox, Christopher Martin (2005). Point-set and point-free topology in constructive set theory (PhD thesis). University ofManchester.

[3] Gambino, Nicolas (2002). Sheaf interpretations for generalised predicative intuitionistic systems (PhD thesis). University ofManchester.

[4] Barthe, Gilles Jacques (1993). Term declaration logic and generalised composita (PhD thesis). University of Manchester.

[5] Koletsos, George (1980). Functional interpretation and β-logic (PhD thesis). University of Manchester.

[6] Väänänen, Jouko Antero (1977). Applications of set theory to generalised quantifiers (PhD thesis). University of Manch-ester.

[7] Peter Aczel at the Mathematics Genealogy Project

[8] http://plato.stanford.edu/entries/nonwellfounded-set-theory/index.html

[9] Aczel, P. (1977). “An Introduction to Inductive Definitions”. Handbook of Mathematical Logic. Studies in Logic and theFoundations of Mathematics 90. pp. 739–201. doi:10.1016/S0049-237X(08)71120-0. ISBN 9780444863881.


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[10] Aczel, P.; Mendler, N. (1989). “A final coalgebra theorem”. Category Theory and Computer Science. Lecture Notes inComputer Science 389. p. 357. doi:10.1007/BFb0018361. ISBN 3-540-51662-X.

[11] Aczel, P. (1980). “Frege Structures and the Notions of Proposition, Truth and Set”. The Kleene Symposium. Studies inLogic and the Foundations ofMathematics 101. pp. 31–32. doi:10.1016/S0049-237X(08)71252-7. ISBN9780444853455.

[12] http://scholar.google.com/scholar?q=peter+aczel Peter Aczel publications in Google Scholar

[13] Peter Aczel’s publications indexed by the DBLP Bibliography Server at the University of Trier

[14] http://www.manchester.ac.uk/research/Peter.Aczel/ Peter Aczel page the University of Manchester

[15] Aczel, Peter (1966). Mathematical problems in logic (DPhil thesis). University of Oxford.(subscription required)

[16] Institute for Advanced Study: A Community of Scholars

[17] http://ndjfl.nd.edu/ Notre Dame Journal of Formal Logic

[18] http://www.journals.elsevier.com/annals-of-pure-and-applied-logic/ Annals of Pure and Applied Logic

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Chapter 11

Tomek Bartoszyński

Tomek Bartoszyński (born May 16, 1957 in Warsaw) is a Polish-American mathematician who works in set theory.He is the son of statistician Robert Bartoszyński.

11.1 Biography

Bartoszyński studied mathematics at the University of Warsaw from 1976 to 1981, and worked there from 1981 to1987. In 1984 he defended his Ph.D. thesis Combinatorial aspects of measure and category; his advisor was WojciechGuzicki.[1] In 2004 he received his habilitation from the Polish Academy of Sciences.From 1986 on he worked in the United States: he taught at University of California in Berkeley and Davis. From 1990to 2006 he was professor (full professor from 1998 on) at Boise State University. In 1990/91 he visited the HebrewUniversity of Jerusalem as a fellow of the Lady Davis foundation, and in 1996/97 he visited the Free University ofBerlin as a Humboldt fellow.Currently he is one of the program directors at the National Science Foundation (NSF), responsible for Combinatorics,Foundations, and Probability.His wife Joanna Kania-Bartoszyńska is the NSF program director for topology and geometric analysis.

11.2 Scientific work

Bartoszyński’s work is mainly concerned with forcing, specifically with applications of forcing to the set theory ofthe real line. He has written about 50 papers in this field, as well as a monograph:

• Tomek Bartoszyński and Haim Judah: Set theory. On the structure of the real line. A K Peters, Ltd., Wellesley,MA, 1995. xii+546 pp. ISBN 1-56881-044-X

11.3 See also

• Cichon’s diagram

• Baire property

11.4 References

[1] Tomek Bartoszyński at the Mathematics Genealogy Project


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11.6 Text and image sources, contributors, and licenses

11.6.1 Text• Alain Badiou Source: https://en.wikipedia.org/wiki/Alain_Badiou?oldid=705274715 Contributors: Gabbe, Andres, Conti, Dysprosia,

Joy, Warofdreams, Rbellin, Bearcat, Fredrik, RedWolf, Aetheling, Diberri, Rj, Mboverload, Gzornenplatz, Andycjp, Spatch, KarolLangner, Phil Sandifer, Ivn~enwiki, Flex, Esperant, D6, David Sneek, Rich Farmbrough, Alex Golub, Lulu of the Lotus-Eaters, Xez-beth, Bender235, MBisanz, Deryck Chan, Jumbuck, VoluntarySlave, Aristides, RyanGerbil10, Gmaxwell, Woohookitty, Radiant!, Man-darax, Xcuref1endx, BD2412, Ketiltrout, Rjwilmsi, Lockley, FlaBot, Moskvax, YurikBot, RobotE, Kordas, RussBot, Billbrock, GaiusCornelius, Nikkimaria, Curpsbot-unicodify, Gaudio, A3ulafia, SmackBot, Roger Davies, Wikikris, TimBentley, Madsanders, Rrburke,Vathek, Salt Yeung, Allisonrung, Michael Rogers, Ohconfucius, Ser Amantio di Nicolao, Giovanni33, John, Santa Sangre, ChristianRoess, Hu12, Twas Now, Fdssdf, Gregbard, Cydebot, Arb, Stephenhowardjones, Thijs!bot, Mephistophilis, 271828182, Jimmyq2305,Okki, Comzero, Morgaledh, Goodlucca, Danny lost, Shogo Kawada, Cleversnail, Skomorokh, Dsp13, Avaya1, MelanieN, Magioladi-tis, Ling.Nut, Waacstats, Lucaas, Skarioffszky, FreddieSpell, Dr.crawboney, Aboutmovies, Dlkj83jdk3883ll, Inwind, VolkovBot, Jim-maths, Zithulele Dlamini, Vanished user ikijeirw34iuaeolaseriffic, Anna Lincoln, Room429, Joelleabirached, BOTijo, Scrawlspacer, Ral-fundflorian, Bobbyperou, Hmwith, Jacques l'Aumône, Tomasboij, SE7, Dlukenelson, Bigdaddy1981, Vojvodaen, Bjorn Martiz, Le vinblanc, Enriquemartino, JL-Bot, JustinBlank, ImageRemovalBot, RS1900, Binksternet, Ark2120, Jpehs, RashersTierney, Niceguyedc,Solar-Wind, DragonBot, Sq178pv, The Audient Void, Tmonzenet, David Šenek, XLinkBot, Jprw, Good Olfactory, Kbdankbot, Addbot,Rachel0898, Saquetin, Lal Salaam, Aryder779, NatanHaasnoot, Vgovind, Favonian, Woland1234, Lightbot, Vitoria666, Brokenjazz, En-bowles, Luckas-bot, Yobot, Andreasmperu, Irønie, Sublirony, Aojohnston, Mahmudmasri, Palacaguina~enwiki, Elnegroeliecer, Xqbot,JimVC3, Gugs redux, Paperoverman, Omnipaedista, Eugene-elgato, FrescoBot, Pilchard7, MaximeMérineau, Eipnvn, Waynechuck, Oli-broman, Trappist the monk, Gingerup, TheMariborchan, YaniaTierra, Beckett00, RjwilmsiBot, DASHBot, EmausBot, And we drown,Hegemonik, Artiquities, Sholomsholom, Stephan Spahn, Nevetsmahtal, Somerwind, Zeusone, ZéroBot, Gracian3, Alex Ling, Jacobisq,Δ, Albinoni67, Donner60, Wagnermusic, JRMcCann, ChuispastonBot, Haigee2007, If Che Married Rosa, Ron9000, Sinistersnowman,Jleecomer, SzMithrandir, Wbm1058, BG19bot, Pellagrina, Mohamed CJ, Rross856, Dldserial, OttawaAC, Ostera65, Badioustudies,Tyrannus Mundi, Picturesque2, Justincheng12345-bot, Anthrophilos, Hérisson de Cloche, Mogism, Cerabot~enwiki, Luciennedepuis,Jonthawk, Ketxus, RaphaelQS, Gubino, Stamptrader, Eb74734, Jon Swift, 123clock, SoSivr, Kebenaran2020, KasparBot, Spaceleagueand Anonymous: 186

• Andreas Blass Source: https://en.wikipedia.org/wiki/Andreas_Blass?oldid=681314022Contributors: Rich Farmbrough, Rjwilmsi, Smack-Bot, Stotr~enwiki, CBM, Cydebot, Ntsimp, SGGH, TAnthony, Magioladitis, Waacstats, David Eppstein, Johnpacklambert, Addbot,Yobot, YMS, Citation bot, Citation bot 1, AvicBot, ZéroBot, Suslindisambiguator, VIAFbot, KasparBot and Anonymous: 4

• Bohuslav Balcar Source: https://en.wikipedia.org/wiki/Bohuslav_Balcar?oldid=659939570 Contributors: SmackBot, Wizardman, Cy-debot, Ntsimp, Waacstats, LBehounek, RogDel, Suslindisambiguator, VIAFbot and KasparBot

• CesareBurali-Forti Source: https://en.wikipedia.org/wiki/Cesare_Burali-Forti?oldid=705570598Contributors: Deb, CharlesMatthews,MathMartin, Peruvianllama, Mellum, Oleg Alexandrov, ABot, Mathbot, SmackBot, Amalas, Pierre de Lyon, Gregbard, Studerby, Storkk,Waacstats, David Eppstein, STBotD, FlagSteward, RogDel, Addbot, Lightbot, Yobot, RedBot, Full-date unlinking bot, Jesse V., Emaus-Bot, Catlemur, Solomon7968, Makecat-bot, VIAFbot, 900mill, Delphenich, Simon Shahin, ArmbrustBot, Hugo Fuxa, KasparBot andAnonymous: 7

• Heinz Bachmann Source: https://en.wikipedia.org/wiki/Heinz_Bachmann?oldid=696572476 Contributors: R.e.b. and Bamyers99• James Earl Baumgartner Source: https://en.wikipedia.org/wiki/James_Earl_Baumgartner?oldid=660162425 Contributors: Aleph4,

Bender235, Oleg Alexandrov, Lockley, R.e.b., Daderot, RayAYang, Racklever, BrownHairedGirl, CBM, Myasuda, Cydebot, Ntsimp,Magioladitis, Connormah, Waacstats, David Eppstein, Kope, SieBot, Addbot, Lightbot, Omnipaedista, RjwilmsiBot, VIAFbot, Kaspar-Bot and Anonymous: 2

• JohnLaneBell Source: https://en.wikipedia.org/wiki/John_Lane_Bell?oldid=693588169Contributors: Rich Farmbrough, Ucucha, Smack-Bot, Afasmit, Colonies Chris, Nick Levine, Saadanis, Gregbard, MainlyTwelve, Acroterion, Magioladitis, Waacstats, Johnpacklambert,Tassedethe, Yobot, Omnipaedista, Tkuvho, Oracleofottawa, RjwilmsiBot, Suslindisambiguator, Johnlbell37, AdsoDaMelk, VIAFbot,KasparBot, Srednuas Lenoroc and Anonymous: 13

• L.E. J. Brouwer Source: https://en.wikipedia.org/wiki/L._E._J._Brouwer?oldid=702661423Contributors: XJaM,WilliamAvery,MichaelHardy, Dominus, Rp, GTBacchus, Med, Charles Matthews, Lfh, David Shay, Jaredwf, Lesonyrra, Jan Lapère, Giftlite, Everyking, Brona,Curps, Yekrats, D3, Pethan, Icairns, TonyW, D6, Zaheen, Bender235, Treborbassett, Haham hanuka, Mdd, Alansohn, KTC, Oleg Alexan-drov, Kzollman, Triddle, Eras-mus, Waldir, Porcher, Wikix, Rjwilmsi, R.e.b., Olessi, FlaBot, JYOuyang, Chobot, YurikBot, 4C~enwiki,Aldux, Tomisti, KnightRider~enwiki, SmackBot, Lestrade, Afasmit, CraigDesjardins, Michael Rogers, Tesseran, Lambiam, Wvbailey,Physis, Hobophobe, CBM, Gregbard, Cydebot, Jasperdoomen, Frostlion, Quibik, Thijs!bot, Epbr123, Dsp13, .anacondabot, Yonidebot,Katharineamy, VolkovBot, NikolaiLobachevsky, Jimmaths, Ontoraul, Duncan.Hull, Insanity Incarnate, E. H.-A. Gerbracht, Katzmik,SieBot, Ekindedeoglu~enwiki, ImageRemovalBot, RS1900, DragonBot, Vegas Bleeds Neon, Universityuser, El bot de la dieta, Certes,FlorisHJ, Johnuniq, Crowsnest, TimothyRias, Addbot, Luckas-bot, Yobot, ArthurBot, Xqbot, Ekwos, Drilnoth, Omnipaedista, FrescoBot,Controle2, Plucas58, RedBot, Full-date unlinking bot, Oracleofottawa, Angelorf, RjwilmsiBot, Suslindisambiguator, Staszek Lem, Tagib,Polisher of Cobwebs, Helpful Pixie Bot, Schojoha, Dexbot, VIAFbot, Crispulop, OccultZone, Peter238, KasparBot, Marcelbeemster andAnonymous: 32

• Paul Bernays Source: https://en.wikipedia.org/wiki/Paul_Bernays?oldid=705704935 Contributors: Zundark, XJaM, Charles Matthews,Hyacinth, Btljs, Giftlite, Rich Farmbrough, Bender235, Sligocki, GeneNygaard, OlegAlexandrov, Sheynhertz-Unbayg, Rjwilmsi, MarSch,Salix alba, R.e.b., Mathbot, SpuriousQ, SmackBot, RayAYang, Amalas, Kylu, Myasuda, Gregbard, Thijs!bot, Waacstats, David Eppstein,Fruits Monster, Daniele.tampieri, Plindenbaum, Alan U. Kennington, Ontoraul, Gian-2, Popopp, Softtest123, Muro Bot, Palnot, Addbot,Download, Luckas-bot, Yobot, Citation bot, Tbvdm, Omnipaedista, SternJacob, Foobarnix, Numericana, ,יניבפור RjwilmsiBot, ZéroBot,Suslindisambiguator, Miszatomic, VIAFbot, Jochen Burghardt, KasparBot, SSTflyer and Anonymous: 15

• Peter Aczel Source: https://en.wikipedia.org/wiki/Peter_Aczel?oldid=705289684 Contributors: Tobias Bergemann, RussBot, Moe Ep-silon, SmackBot, SMasters, Gregbard, Magioladitis, Duncan.Hull, Y, Iohannes Animosus, Addbot, Uncia, Yobot, AnomieBOT, Omni-paedista, GainLine, LucienBOT, Foobarnix, Jdapayne, John of Reading, ZéroBot, ChrisGualtieri, Dexbot and Liz

• Tomek Bartoszyński Source: https://en.wikipedia.org/wiki/Tomek_Bartoszy%C5%84ski?oldid=664579141 Contributors: Aleph4, D6,Kbdank71, Lockley, SmackBot, Stotr~enwiki, Cydebot, Ntsimp, Wlod, Waacstats, Johnpacklambert, Martarius, Yobot, RjwilmsiBot,EmausBot, Set theorist, VIAFbot, Liz, KasparBot and Anonymous: 2

Page 39: Set Theorists 1


11.6.2 Images• File:Alain_Badiou-2.jpg Source: https://upload.wikimedia.org/wikipedia/commons/8/89/Alain_Badiou-2.jpg License: CC BY-SA 3.0

Contributors: Own work Original artist: Keffieh67• File:Ambox_important.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b4/Ambox_important.svg License: Public do-

main Contributors: Own work, based off of Image:Ambox scales.svg Original artist: Dsmurat (talk · contribs)• File:AndreasBlass.jpg Source: https://upload.wikimedia.org/wikipedia/commons/a/a2/AndreasBlass.jpg License: GFDL Contributors:

Own work Original artist: Stotr• File:Badiou-an_original_drawing.jpg Source: https://upload.wikimedia.org/wikipedia/en/9/9a/Badiou-an_original_drawing.jpg Li-

cense: Public domain Contributors: ? Original artist: ?• File:BuraliForti1.jpg Source: https://upload.wikimedia.org/wikipedia/commons/a/a2/BuraliForti1.jpg License: Public domain Contrib-

utors: http://www.s9.com/images/portraits/4349_Burali-Forti-Cesare.jpg Original artist: Unknown<a href='//www.wikidata.org/wiki/Q4233718' title='wikidata:Q4233718'><img alt='wikidata:Q4233718' src='https://upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/20px-Wikidata-logo.svg.png' width='20' height='11' srcset='https://upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/30px-Wikidata-logo.svg.png 1.5x, https://upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/40px-Wikidata-logo.svg.png 2x' data-file-width='1050' data-file-height='590' /></a>

• File:Commons-logo.svg Source: https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg License: CC-BY-SA-3.0Contrib-utors: ? Original artist: ?

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• File:Flag_of_the_Czech_Republic.svg Source: https://upload.wikimedia.org/wikipedia/commons/c/cb/Flag_of_the_Czech_Republic.svg License: Public domain Contributors:

• -xfi-'s file• -xfi-'s code• Zirland’s codes of colors

Original artist:(of code): SVG version by cs:-xfi-.

• File:Flag_of_the_United_States.svg Source: https://upload.wikimedia.org/wikipedia/en/a/a4/Flag_of_the_United_States.svg License:PD Contributors: ? Original artist: ?

• File:James_Baumgartner.jpeg Source: https://upload.wikimedia.org/wikipedia/commons/a/a9/James_Baumgartner.jpeg License: CCBY-SA 2.0 Contributors: http://owpdb.mfo.de/detail?photo_id=4770 Original artist: Bergman, George M.

• File:Loudspeaker.svg Source: https://upload.wikimedia.org/wikipedia/commons/8/8a/Loudspeaker.svg License: Public domain Con-tributors: New version of Image:Loudspeaker.png, by AzaToth and compressed by Hautala Original artist: Nethac DIU, waves correctedby Zoid

• File:Luitzen_Egbertus_Jan_Brouwer.jpeg Source: https://upload.wikimedia.org/wikipedia/en/6/6c/Luitzen_Egbertus_Jan_Brouwer.jpeg License: Fair use Contributors: http://www-history.mcs.st-andrews.ac.uk/Biographies/Brouwer.html Original artist: Unknown

• File:NewtonDetail.jpg Source: https://upload.wikimedia.org/wikipedia/commons/3/35/NewtonDetail.jpg License: Public domain Con-tributors: Transferred from en.wikipedia to Commons by Sreejithk2000 using CommonsHelper. Original artist: The original uploaderwas Trovatore at English Wikipedia

• File:PaulBernays_1949_MFO.jpg Source: https://upload.wikimedia.org/wikipedia/commons/6/68/PaulBernays_1949_MFO.jpg Li-cense: CC BY-SA 2.0 de Contributors: Own work Original artist: Author of original file unknown; extraction by Jochen Burghardt

• File:Racine_carrée_bleue.svg Source: https://upload.wikimedia.org/wikipedia/commons/1/1f/Racine_carr%C3%A9e_bleue.svgLicense:LGPL Contributors: Image:Nuvola apps edu mathematics-p.svg Original artist: historicair 17:50, 4 June 2007 (UTC)

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11.6.3 Content license• Creative Commons Attribution-Share Alike 3.0