A Venn diagram illustrating the intersection of two sets.

Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.

The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.

Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.

History[link]

Georg Cantor

Mathematical topics typically emerge and evolve through interactions among many researchers. Set theory, however, was founded by a single paper in 1874 by Georg Cantor: "On a Characteristic Property of All Real Algebraic Numbers".^[1][2]

Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, mathematicians had struggled with the concept of infinity. Especially notable is the work of Bernard Bolzano in the first half of the 19th century.^[3] Modern understanding of infinity began in 1867-71, with Cantor's work on number theory. An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's thinking and culminated in Cantor's 1874 paper.

Cantor's work initially polarized the mathematicians of his day. While Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. Cantorian set theory eventually became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his proof that there are more real numbers than integers, and the "infinity of infinities" ("Cantor's paradise") resulting from the power set operation. This utility of set theory led to the article "Mengenlehre" contributed in 1898 by Arthur Schoenflies to Klein's encyclopedia.

The next wave of excitement in set theory came around 1900, when it was discovered that Cantorian set theory gave rise to several contradictions, called antinomies or paradoxes. Bertrand Russell and Ernst Zermelo independently found the simplest and best known paradox, now called Russell's paradox: consider "the set of all sets that are not members of themselves", which leads to a contradiction since it must be a member of itself, and not a member of itself. In 1899 Cantor had himself posed the question "What is the cardinal number of the set of all sets?", and obtained a related paradox. Russell used his paradox as a theme in his 1903 review of continental mathematics in his Principles of Mathematics.

The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment. The work of Zermelo in 1908 and Abraham Fraenkel in 1922 resulted in the set of axioms ZFC, which became the canonical^{[dubious – discuss]} axioms for set theory. The work of analysts such as Henri Lebesgue demonstrated the great mathematical utility of set theory, which has since become woven into the fabric of modern mathematics. Set theory is commonly used as a foundational system, although in some areas category theory is thought to be a preferred foundation.

Basic concepts[link]

Set theory begins with a fundamental binary relation between an object o and a set A. If o is a member (or element) of A, write o ∈ A. Since sets are objects, the membership relation can relate sets as well.

A derived binary relation between two sets is the subset relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, denoted A ⊆ B. For example, {1,2} is a subset of {1,2,3} , but {1,4} is not. From this definition, it is clear that a set is a subset of itself; in cases where one wishes to avoid this, the term proper subset is defined to exclude this possibility.

Just as arithmetic features binary operations on numbers, set theory features binary operations on sets. The:

• Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both. The union of {1, 2, 3} and {2, 3, 4} is the set {1, 2, 3, 4} .
• Intersection of the sets A and B, denoted A ∩ B, is the set of all objects that are members of both A and B. The intersection of {1, 2, 3} and {2, 3, 4} is the set {2, 3} .
• Set difference of U and A, denoted U \ A is the set of all members of U that are not members of A. The set difference {1,2,3} \ {2,3,4} is {1} , while, conversely, the set difference {2,3,4} \ {1,2,3} is {4} . When A is a subset of U, the set difference U \ A is also called the complement of A in U. In this case, if the choice of U is clear from the context, the notation A^c is sometimes used instead of U \ A, particularly if U is a universal set as in the study of Venn diagrams.
• Symmetric difference of sets A and B is the set of all objects that are a member of exactly one of A and B (elements which are in one of the sets, but not in both). For instance, for the sets {1,2,3} and {2,3,4} , the symmetric difference set is {1,4} . It is the set difference of the union and the intersection, (A ∪ B) \ (A ∩ B).
• Cartesian product of A and B, denoted A × B, is the set whose members are all possible ordered pairs (a,b) where a is a member of A and b is a member of B. The cartesian product of {1, 2} and {red, white} is {(1, red), (1, white), (2, red), (2, white)}.
• Power set of a set A is the set whose members are all possible subsets of A. For example, the power set of {1, 2} is { {}, {1}, {2}, {1,2} } .

Some basic sets of central importance are the empty set (the unique set containing no elements), the set of natural numbers, and the set of real numbers.

Some ontology[link]

An initial segment of the von Neumann hierarchy.

A set is pure if all of its members are sets, all members of its members are sets, and so on. For example, the set {{}} containing only the empty set is a nonempty pure set. In modern set theory, it is common to restrict attention to the von Neumann universe of pure sets, and many systems of axiomatic set theory are designed to axiomatize the pure sets only. There are many technical advantages to this restriction, and little generality is lost, since essentially all mathematical concepts can be modeled by pure sets. Sets in the von Neumann universe are organized into a cumulative hierarchy, based on how deeply their members, members of members, etc. are nested. Each set in this hierarchy is assigned (by transfinite recursion) an ordinal number α, known as its rank. The rank of a pure set X is defined to be one more than the least upper bound of the ranks of all members of X. For example, the empty set is assigned rank 0, while the set {{}} containing only the empty set is assigned rank 1. For each ordinal α, the set V_α is defined to consist of all pure sets with rank less than α. The entire von Neumann universe is denoted V.

Axiomatic set theory[link]

Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using Venn diagrams. The intuitive approach tacitly assumes that a set may be formed from the class of all objects satisfying any particular defining condition. This assumption gives rise to paradoxes, the simplest and best known of which are Russell's paradox and the Burali-Forti paradox. Axiomatic set theory was originally devised to rid set theory of such paradoxes.^[4]

The most widely studied systems of axiomatic set theory imply that all sets form a cumulative hierarchy. Such systems come in two flavors, those whose ontology consists of:

• Sets alone. This includes the most common axiomatic set theory, Zermelo–Fraenkel set theory (ZFC), which includes the axiom of choice. Fragments of ZFC include:
  • Zermelo set theory, which replaces the axiom schema of replacement with that of separation;
  • General set theory, a small fragment of Zermelo set theory sufficient for the Peano axioms and finite sets;
  • Kripke-Platek set theory, which omits the axioms of infinity, powerset, and choice, and weakens the axiom schemata of separation and replacement.
• Sets and proper classes. This includes Von Neumann-Bernays-Gödel set theory, which has the same strength as ZFC for theorems about sets alone, and Morse-Kelley set theory, which is stronger than ZFC.

The above systems can be modified to allow urelements, objects that can be members of sets but that are not themselves sets and do not have any members.

The systems of New Foundations NFU (allowing urelements) and NF (lacking them) are not based on a cumulative hierarchy. NF and NFU include a "set of everything," relative to which every set has a complement. In these systems urelements matter, because NF, but not NFU, produces sets for which the axiom of choice does not hold.

Systems of constructive set theory, such as CST, CZF, and IZF, embed their set axioms in intuitionistic logic instead of first order logic. Yet other systems accept standard first order logic but feature a nonstandard membership relation. These include rough set theory and fuzzy set theory, in which the value of an atomic formula embodying the membership relation is not simply True or False. The Boolean-valued models of ZFC are a related subject.

An enrichment of ZFC called Internal Set Theory was proposed by Edward Nelson in 1977.

Applications[link]

Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as graphs, manifolds, rings, and vector spaces can all be defined as sets satisfying various (axiomatic) properties. Equivalence and order relations are ubiquitous in mathematics, and the theory of mathematical relations can be described in set theory.

Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of Principia Mathematica, it has been claimed that most or even all mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using first or second order logic. For example, properties of the natural and real numbers can be derived within set theory, as each number system can be identified with a set of equivalence classes under a suitable equivalence relation whose field is some infinite set.

Set theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete mathematics is likewise uncontroversial; mathematicians accept that (in principle) theorems in these areas can be derived from the relevant definitions and the axioms of set theory. Few full derivations of complex mathematical theorems from set theory have been formally verified, however, because such formal derivations are often much longer than the natural language proofs mathematicians commonly present. One verification project, Metamath, includes derivations of more than 10,000 theorems starting from the ZFC axioms and using first order logic.

Areas of study[link]

Set theory is a major area of research in mathematics, with many interrelated subfields.

Combinatorial set theory[link]

Combinatorial set theory concerns extensions of finite combinatorics to infinite sets. This includes the study of cardinal arithmetic and the study of extensions of Ramsey's theorem such as the Erdős–Rado theorem.

Descriptive set theory[link]

Descriptive set theory is the study of subsets of the real line and, more generally, subsets of Polish spaces. It begins with the study of pointclasses in the Borel hierarchy and extends to the study of more complex hierarchies such as the projective hierarchy and the Wadge hierarchy. Many properties of Borel sets can be established in ZFC, but proving these properties hold for more complicated sets requires additional axioms related to determinacy and large cardinals.

The field of effective descriptive set theory is between set theory and recursion theory. It includes the study of lightface pointclasses, and is closely related to hyperarithmetical theory. In many cases, results of classical descriptive set theory have effective versions; in some cases, new results are obtained by proving the effective version first and then extending ("relativizing") it to make it more broadly applicable.

A recent area of research concerns Borel equivalence relations and more complicated definable equivalence relations. This has important applications to the study of invariants in many fields of mathematics.

Fuzzy set theory[link]

In set theory as Cantor defined and Zermelo and Fraenkel axiomatized, an object is either a member of a set or not. In fuzzy set theory this condition was relaxed by Lotfi A. Zadeh so an object has a degree of membership in a set, as number between 0 and 1. For example, the degree of membership of a person in the set of "tall people" is more flexible than a simple yes or no answer and can be a real number such as 0.75.

Inner model theory[link]

An inner model of Zermelo–Fraenkel set theory (ZF) is a transitive class that includes all the ordinals and satisfies all the axioms of ZF. The canonical example is the constructible universe L developed by Gödel. One reason that the study of inner models is of interest is that it can be used to prove consistency results. For example, it can be shown that regardless whether a model V of ZF satisfies the continuum hypothesis or the axiom of choice, the inner model L constructed inside the original model will satisfy both the generalized continuum hypothesis and the axiom of choice. Thus the assumption that ZF is consistent (has any model whatsoever) implies that ZF together with these two principles is consistent.

The study of inner models is common in the study of determinacy and large cardinals, especially when considering axioms such as the axiom of determinacy that contradict the axiom of choice. Even if a fixed model of set theory satisfies the axiom of choice, it is possible for an inner model to fail to satisfy the axiom of choice. For example, the existence of sufficiently large cardinals implies that there is an inner model satisfying the axiom of determinacy (and thus not satisfying the axiom of choice).^[5]

Large cardinals[link]

A large cardinal is a cardinal number with an extra property. Many such properties are studied, including inaccessible cardinals, measurable cardinals, and many more. These properties typically imply the cardinal number must be very large, with the existence of a cardinal with the specified property unprovable in Zermelo-Fraenkel set theory.

Determinacy[link]

Determinacy refers to the fact that, under appropriate assumptions, certain two-player games of perfect information are determined from the start in the sense that one player must have a winning strategy. The existence of these strategies has important consequences in descriptive set theory, as the assumption that a broader class of games is determined often implies that a broader class of sets will have a topological property. The axiom of determinacy (AD) is an important object of study; although incompatible with the axiom of choice, AD implies that all subsets of the real line are well behaved (in particular, measurable and with the perfect set property). AD can be used to prove that the Wadge degrees have an elegant structure.

Forcing[link]

Paul Cohen invented the method of forcing while searching for a model of ZFC in which the axiom of choice or the continuum hypothesis fails. Forcing adjoins to some given model of set theory additional sets in order to create a larger model with properties determined (i.e. "forced") by the construction and the original model. For example, Cohen's construction adjoins additional subsets of the natural numbers without changing any of the cardinal numbers of the original model. Forcing is also one of two methods for proving relative consistency by finitistic methods, the other method being Boolean-valued models.

Cardinal invariants[link]

A cardinal invariant is a property of the real line measured by a cardinal number. For example, a well-studied invariant is the smallest cardinality of a collection of meagre sets of reals whose union is the entire real line. These are invariants in the sense that any two isomorphic models of set theory must give the same cardinal for each invariant. Many cardinal invariants have been studied, and the relationships between them are often complex and related to axioms of set theory.

Set-theoretic topology[link]

Set-theoretic topology studies questions of general topology that are set-theoretic in nature or that require advanced methods of set theory for their solution. Many of these theorems are independent of ZFC, requiring stronger axioms for their proof. A famous problem is the normal Moore space question, a question in general topology that was the subject of intense research. The answer to the normal Moore space question was eventually proved to be independent of ZFC.

Objections to set theory as a foundation for mathematics[link]

From set theory's inception, some mathematicians objected to it as a foundation for mathematics. The most common objection to set theory, one Kronecker voiced in set theory's earliest years, starts from the constructivist view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both in naive and in axiomatic set theory, introduces into mathematics methods and objects that are not computable even in principle. Ludwig Wittgenstein questioned the way Zermelo–Fraenkel set theory handled infinities.^{[citation needed]} Wittgenstein's views about the foundations of mathematics were later criticised by Georg Kreisel and Paul Bernays, and closely investigated by Crispin Wright, among others.

Category theorists have proposed topos theory as an alternative to traditional axiomatic set theory. Topos theory can interpret various alternatives to that theory, such as constructivism, finite set theory, and computable set theory.^[6]

See also[link]

	Set theory portal
	Mathematics portal

• Category theory
• List of set theory topics
• Musical set theory concerns the application of combinatorics and group theory to music; beyond the fact that it uses finite sets it has nothing to do with mathematical set theory of any kind. In the last two decades, transformational theory in music has taken the concepts of mathematical set theory more rigorously (see Lewin 1987).
• Relational model - Borrows from Set Theory.

Notes[link]

Further reading[link]

• Devlin, Keith, (2nd ed.) 1993. The Joy of Sets. Springer Verlag, ISBN 0-387-94094-4
• Ferreirós, Jose, 2007 (1999). Labyrinth of Thought: A history of set theory and its role in modern mathematics. Basel, Birkhäuser. ISBN 978-3-7643-8349-7
• Johnson, Philip, 1972. A History of Set Theory. Prindle, Weber & Schmidt ISBN 0-87150-154-6
• Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. North-Holland, ISBN 0-444-85401-0.
• Tiles, Mary, 2004 (1989). The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise. Dover Publications.

External links[link]

Wikibooks has a book on the topic of Set Theory

Wikibooks has a book on the topic of Discrete mathematics/Set theory

• Foreman, M., Akihiro Kanamori, eds. Handbook of Set Theory. 3 vols., 2010. Each chapter surveys some aspect of contemporary research in set theory. Does not cover established elementary set theory, on which see Devlin (1993).
• Arthur Schoenflies (1898) Mengenlehre in Klein's encyclopedia.

Alain Badiou

Alain Badiou. Fnac Montparnasse (Paris). February 5th, 2010
Born	(1937-01-17) 17 January 1937 (age 75) Rabat, French Morocco
Era	Contemporary philosophy
Region	French philosophy
School	Marxism Continental philosophy
Main interests	Set Theory, Mathematics, Metapolitics, Ontology, Marxism
Notable ideas	Événement (Event), ontologie du multiple (ontology of the multiple & ontology is mathematics), Un n'est pas ("The One is Not")
Influenced by Plato, Marx, Cantor, Albert Lautman, Mao Zedong, Lacan, Althusser, Paul Cohen, Sartre, Deleuze, Lyotard, Hegel, Mallarmé, Beckett, Fernando Pessoa, Sylvain Lazarus
Influenced Slavoj Žižek, Peter Hallward, Simon Critchley, Ray Brassier, Sylvain Lazarus, Jason Barker, Quentin Meillassoux, Bruno Bosteels, Alberto Toscano, Oliver Feltham, Justin Clemens, Nina Power

Alain Badiou (French pronunciation: [alɛ̃ badju]  (listen) (help·info); born 17 January 1937 in Rabat, Morocco) is a French philosopher, professor at European Graduate School, formerly chair of Philosophy at the École Normale Supérieure (ENS).^[1]

Badiou has written about the concepts of being, truth and the subject in a way that, he claims, is neither postmodern nor simply a repetition of modernity. Politically, Badiou is committed to the far left, and to the communist tradition.

Slavoj Žižek has written of Badiou that he is "a figure like Plato or Hegel walk[ing] here among us".^[2]

Biography[link]

Badiou was a student at the Lycée Louis-Le-Grand and then the Ecole Normale Supérieure (1957–1961). He taught at the lycée in Reims from 1963 where he became a close friend of fellow playwright (and philosopher) François Regnault,^[3] and published a couple of novels before moving to the University of Paris VIII (Vincennes-Saint Denis) in 1969.^[4] Badiou was politically active very early on, and was one of the founding members of the Unified Socialist Party (PSU). The PSU was particularly active in the struggle for the decolonization of Algeria. He wrote his first novel, Almagestes, in 1964. In 1967 he joined a study group organized by Louis Althusser, became increasingly influenced by Jacques Lacan and became a member of the editorial board of Cahiers pour l’Analyse.^[4] By then he "already had a solid grounding in mathematics and logic (along with Lacanian theory)",^[4] and his own two contributions to the pages of Cahiers "anticipate many of the distinctive concerns of his later philosophy".^[4]

The student uprisings of May 1968 reinforced Badiou's commitment to the far Left, and he participated in increasingly militant 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 a fair number of young people".^[5] During this time, Badiou joined the faculty of the newly founded University of Paris VIII/Vincennes-Saint Denis which was a bastion of counter-cultural thought. There he engaged in fierce intellectual debates with fellow professors Gilles Deleuze and Jean-François Lyotard, whose philosophical works he considered unhealthy deviations from the Althusserian program of a scientific Marxism.

In the 1980s, as both Althusserian Marxism and Lacanian psychoanalysis went into decline (with Lacan dead and Althusser in an asylum), Badiou published more technical and abstract philosophical works, such as Théorie du sujet (1982), and his magnum opus, Being and Event (1988). Nonetheless, Badiou has never renounced Althusser or Lacan, and sympathetic references to Marxism and psychoanalysis are not uncommon in his more recent works (most notably Petit panthéon portatif/Pocket Pantheon).^[6][7]

He took up his current position at the ENS in 1999. He is also associated with a number of other institutions, such as the 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, according to the French Wikipedia article (linked to in the previous sentence). 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 in American and English periodicals, such as Lacanian Ink, New Left Review, Radical Philosophy, Cosmos and History [1] and Parrhesia. Unusually for a contemporary European philosopher his work is increasingly being taken up by militants in movements of the poor in countries like India, the Democratic Republic of Congo and South Africa where he is often read together with Frantz Fanon.^{[citation needed]}

Lately Badiou got into a fierce controversy within the confines of Parisian intellectual life. It started in 2005 with the publication of his "Circonstances 3: Portées du mot 'juif'" – The Uses of the Word "Jew".^[8] This book generated a strong response with calls of Badiou being labelled Anti-Semitic. The wrangling became a cause célèbre with articles going back and forth in the French newspaper Le Monde and in the cultural journal "Les temps modernes." Another philosopher, Jean-Claude Milner, has accused Badiou of Anti-Semitism.^[9]

Key concepts[link]

Badiou makes repeated use of several concepts throughout his philosophy. One of the aims of his thought is to show that his categories of truth are useful for any type of philosophical critique. Therefore, he uses them to interrogate art and history as well as ontology and scientific discovery. Johannes Thumfart argues that Badiou's philosophy can be regarded as a contemporary reinterpretation of Platonism.^[10]

Conditions[link]

According to Badiou, philosophy is suspended from four conditions (art, love, politics, and science), each of them fully independent "truth procedures." (For Badiou's notion of truth procedures, see below.) Badiou consistently maintains throughout his work (but most systematically in Manifesto for Philosophy) that philosophy must avoid the temptation to suture itself (that is, to hand over its entire intellectual effort) to any of these independent truth procedures. When philosophy does suture itself to one of its conditions (and Badiou argues that the history of philosophy during the nineteenth and twentieth centuries is primarily a history of sutures), what results is a philosophical "disaster." Consequently, philosophy is, according to Badiou, a thinking of the compossibility of the several truth procedures, whether this is undertaken through the investigation of the intersections between distinct truth procedures (the intersection of art and love in the novel, for instance), or whether this is undertaken through the more traditionally philosophical work of addressing categories like truth or the subject (concepts that are, as concepts, external to the individual truth procedures, though they are functionally operative in the truth procedures themselves). For Badiou, when philosophy addresses the four truth procedures in a genuinely philosophical manner, rather than through a suturing abandonment 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 own terms, "truth procedures" (i.e., they produce truths as they are pursued), it is only philosophy that can speak of the several truth procedures as truth procedures. (The lover, for instance, does not think of her love as a question of truth, but simply and rightly as a question of love. Only the philosopher sees in the true lover's love the unfolding of a truth.) Badiou has a very rigorous notion of truth, one that is strongly against the grain of much of contemporary European 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 are constructed through processes. Badiou's theory of truth, exposited throughout his work, accomplishes this strange mixture by uncoupling invariance and self-evidence (such that invariance does not imply self-evidence), as well as by 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 always the case, it passes unnoticed unless there is a rupture in the laws of being and appearance, during which the truth in 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 introduce the truth by naming it into worldly situations. For Badiou, it is by positioning oneself to the truth of an event that a human animal becomes a subject; subjectivity is not an inherent human trait. According to a process or procedure that subsequently unfolds only if those who subject themselves to the glimpsed truth continue faithful in the work of announcing the truth in question, genuine knowledge is produced (knowledge often appears in Badiou's work under 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 to what may yet be produced as faithful subjects of the event produce further knowledge. According to Badiou, truth procedures proceed to infinity, such that faith (fidelity) outstrips knowledge. (Badiou, following both Lacan and Heidegger, distances truth from knowledge.) The dominating ideology of the day, which Badiou terms "democratic materialism," denies the existence of truth and only recognize "bodies" and "languages." Badiou proposes a turn towards "dialectical materialism," which recognizes that there are only bodies and languages, except there are also truths.

Inaesthetic[link]

In Handbook of Inaesthetics Badiou coins the phrase "inaesthetic" to refer to a concept of artistic creation that denies "the reflection/object relation". Reacting against the idea of mimesis, or poetic reflection of "nature", Badiou claims that art is "immanent" and "singular". It is immanent in the sense that its truth is given in its immediacy in a given work of art, and singular in that its truth is found in art and art alone. His view of the link between philosophy and art is tied into the motif of pedagogy, which he claims functions so as to "arrange the forms of knowledge in a way that some truth may come to pierce a hole in them". He develops these ideas with examples from the prose of Samuel Beckett and the poetry of Stéphane Mallarmé and Fernando Pessoa (who he argues has developed a body of work that philosophy is currently incapable of incorporating), among others.

[edit] Introduction to Being and Event

The major propositions of Badiou's philosophy all find their basis in Being and Event, in which he continues his attempt (which he began in Théorie du sujet) to reconcile a notion of the subject with ontology, and in particular post-structuralist and constructivist ontologies.^[11] A frequent criticism of post-structuralist work is that it prohibits, through its fixation on semiotics and language, any notion of a subject. Badiou's work is, by his own admission,^[12] an attempt to break out of contemporary philosophy's fixation upon language, which he sees almost as a straitjacket. This effort leads him, in Being and Event, to combine rigorous mathematical formulae with his readings of poets such as Mallarmé and 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, meaning that his work had been only slowly taken up.^[13] Being and Event offers an example of this slow uptake, in fact: it was translated 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 rupture which characterizes the event are thought in terms of set theory, and specifically Zermelo–Fraenkel set theory (with the axiom of choice), to which Badiou accords a fundamental role in a manner quite distinct from the majority of either mathematicians or philosophers.

Mathematics as ontology[link]

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For Badiou the problem which the Greek tradition of philosophy has faced and never satisfactorily dealt with is that while 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 is not. 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 of the multiple'. Set theory does not operate in terms of definite individual elements in groupings but only functions insofar as what belongs to a set is of the same relation as that set (that is, another set too). What separates sets out therefore is not an existential positive proposition, but other multiples whose properties validate its presentation; which is to say their structural relation. The structure of being thus secures the regime of the count-as-one. So if one is to think of a set – for instance, the set of people, or humanity – as counting as one the elements which belong to that set, it can then secure the multiple (the multiplicities of humans) as one consistent concept (humanity), but only in terms of what does not belong to that set. What is, in following, crucial for Badiou is that the structural form of the count-as-one, which makes multiplicities thinkable, implies that the proper name of being does not belong to an element as such (an original 'one'), but rather the void set (written Ø), the set to which nothing (not even the void set itself) belongs. It may help to understand the concept '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' does not contradict the non-being of the one can be understood by considering the multiple nature of terminology: for there to be a term without there also being a system of terminology, within which the difference between terms gives context and meaning to any one term, does not coincide with what is understood by 'terminology', which is precisely difference (thus multiplicity) conditioning meaning. Since the idea of conceiving of a term without meaning does not compute, the count-as-one is a structural effect or a situational operation and not an event of truth. Multiples which are 'composed' or 'consistent' are count-effects; inconsistent multiplicity is 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 significantly this use means that (as with set theory) there is a strict prohibition on self-belonging; a set cannot contain or belong to itself. Russell's paradox famously ruled that possibility out of formal logic. (This paradox can be thought through in terms of a 'list of lists that do not contain themselves': if such a list does not write itself on the list the property is incomplete, as there will be one missing; if it does, it is no longer a list that does not contain itself.) So too does the axiom of foundation – or to give an alternative name the axiom of regularity – enact such a prohibition (cf. p. 190 in Being and Event). (This axiom states that all sets contain an element for which only the void [empty] set names what is common to both the set and its element.) Badiou's philosophy draws two major implications from this prohibition. Firstly, it secures the inexistence of the 'one': there cannot be a grand overarching set, and thus it is fallacious to conceive of a grand cosmos, a whole Nature, or a Being of God. Badiou is therefore – against Cantor, from whom he draws heavily – staunchly atheist. However, secondly, this prohibition prompts him to introduce the event. Because, according to Badiou, the axiom of foundation 'founds' all sets in the void, it ties all being to the historico-social situation of the multiplicities of de-centred sets – thereby effacing the positivity of subjective action, or an entirely 'new' occurrence. And whilst this is acceptable 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 one possibility remaining: that ontology can say nothing about the event.

The event and the subject[link]

The principle of the event is where Badiou diverges from the majority of late twentieth century philosophy and social thought, and in particular the likes of Foucault, Butler, Lacan and Deleuze, among others. In short, it represents that which is outside of ontology. Badiou's problem here is, unsurprisingly, the question of how to 'make use' of that which cannot be discerned. But it is a problem he views as vital, because if one constructs the world only from that which can be discerned and therefore given a name, it results in either the destitution of subjectivity and the removal of the subject from ontology (the criticism continually leveled at Foucault's discursive universe), or the Panglossian solution of 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 an indiscernible element existing extrinsically to the situation of ontology. He employs the strategy of the mathematician Paul J. Cohen, using what are called the conditions of sets. These conditions are thought of in terms of domination, a domination being that which defines a set. (If one takes, in binary language, the set with the condition 'items marked only with ones', any item marked with zero negates the property of the set. The condition which has only ones is thus dominated by any condition which has zeros in it [cf. p. 367-71 in Being and Event].) Badiou reasons using these conditions that every discernible (nameable or constructible) set is dominated by the conditions which don't possess the property that makes it discernible as a set. (The property 'one' is always dominated by 'not one'.) These sets are, in line with constructible ontology, relative to one's being-in-the-world and one's being in language (where sets and concepts, such as the concept 'humanity', get their names). However, he continues, the dominations themselves are, whilst being relative concepts, not necessarily intrinsic to language and constructible thought; rather one can axiomatically define a domination – in the terms of mathematical ontology – as a set of conditions such that any condition outside the domination is dominated by at least one term inside the domination. One does not necessarily need 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 constructible universe of language, by a process Cohen calls forcing. And he concludes in following that while ontology can mark out 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 thus nominate, and give name to, the undecidable event. Badiou thereby marks out a philosophy by which to refute the apparent 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 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', which in 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 the present situation and the evental [sic] (his translators' neologism) rupture. So too does love have this characteristic of becoming 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 truth comes 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 physics by all the chance encounters that are named in subjects such as Descartes or Newton. How could he, with the 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 modern science; that is, the supplementation of his situation with the indiscernible and unfinishable part that one has to name ‘rational physics’?

Badiou, whilst keen to stress the non-equivalence between politics and philosophy, thus finds his political approach – one of activism, militancy, and scepticism of parliamentary-democratic process – backed up by his philosophy based around singular, situated truths, and potential revolutions.

L'Organisation Politique[link]

Alain Badiou is a founding member (along with Natacha Michel and Sylvain Lazarus) of the militant French political organisation L'Organisation Politique, which was active from 1985 until it disbanded in 2007.^[14] It called itself a post-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 stressed the importance of developing political prescriptions concerning undocumented migrants (les sans papiers), stressing that they must be conceived primarily as workers and not immigrants.^[15]

Works[link]

English translations[link]

Journals[link]

• The International Journal of Badiou Studies
• "The Cultural Revolution: The Last Revolution?", transl. by Bruno Bosteels; positions: asia critique, Volume 13, 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 assistance of Lorenzo Chiesa and Nina Power; positions: asia critique, Volume 13, Issue 3, Winter 2005; (Durham: Duke University 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, Winter 2005; (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): ISSN 1067-9847

DVD[link]

Lectures[link]

Further reading[link]

Notes[link]

External links[link]

Wikimedia Commons has media related to: Alain Badiou

Wikiquote has a collection of quotations related to: Alain Badiou

• Alain Badiou Faculty Page at European Graduate School
• Alain Badiou Bibliography at Lacan Dot Com
• What is a philosophical Institution? or: Address, Transmission, Inscription. Cosmos and History: The Journal of Natural and Social Philosophy, Vol 2, No 1-2 (2006)

Critical Opinions[link]

• Alain Badiou's "Politics of Emancipation": A Communism Locked Within the Confines of the Bourgeois World by Raymond Lotta, Nayi Duniya, and K. J. A.
• On Alain Badiou and Logiques des mondes by Slavoj Zizek
• The Marxist hypothesis: a response to Alain Badiou's "communist hypothesis" by Chris Cutrone