Everything you needed to know about forcing, but were afraid to ask Alain Badiou

Hat tip to Fabio at Hypertiling and Tzuchien Tho for bringing this to my attention. Wish I could be there. If the reason you can't attend this magnificent, London-based workshop is that you'll be in Mexico City this May, stay tuned...

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A Set Theory Postgraduate Workshop for Readers of Being and Event or: Everything You Needed to Know about Forcing but were Afraid to Ask Alain Badiou

24 May, 31 May and 7 June 2011 in Room G03, 28 Russell Square

As part of its ongoing Seminar Series on Mathematics for the Humanities and Cultural Studies, the London Consortium will hold a series of three postgraduate students workshops that aims to discuss and provide a quick overview of some of the mathematics that needs to be known in order to follow the use of set theory
in Alain Badiou’s Being and Event [L'Être et l'Événement]. The main focus of these ‘bootcamps’ will be on introducing and discussing, in a friendly manner, the technicalities concerning the “mathematical bulwarks” mentioned in Badiou’s philosophical masterpiece:
(1) the Zermelo-Fraenkel Axioms of Set Theory
(2) the Theory of Ordinal and Cardinal Numbers
(3) Kurt Gödel and Paul Cohen’s work on Consistency and Independence.

Particular attention will be paid towards providing a sufficiently detailed, rigorous and clear explication of Paul Cohen’s technique of forcing and generic models, a mathematical result that Peter Hallward has called “the single most important postulate” in the whole of Being and Event. If time permits, we might also be touching on some recent mathematical developments in the field as well as trying to understand how Badiou contributes towards the contemporary re-intervention of mathematical thinking into philosophy.

The workshops will be held in room G03 at 28 Russell Square, Bloomsbury, London. The intended audience are postgraduate students or researchers who are interested in understanding Badiou’s philosophy but who lack the mathematical background. There are no assigned compulsory readings for the bootcamps,
and there are no pre-requisites save for some minimal familiarity with mathematics at the pre-university level. The sessions are free and open to the public but please register by sending your name, email and affiliation to forcing.badiou@gmx.com so as to give us an idea of the numbers since the classroom size, unfortunately, will be limited.

Workshop Convener: Burhanuddin Baki

Schedule and List of Topics

Session I (Tuesday, 24 May)

2-4pm – Basic Mathematics and Short Introduction to Forcing
Short Introduction to Badiou’s Being and Event; Basic Arithmetic, Abstract Algebra and First-Order Logic;
Idea of Mathematical Proof; Induction; Naïve Set Theory, Cantor’s Theorem and Russell’s Paradox

6-8pm – Zermelo-Fraenkel Set Theory
Zermelo-Fraenkel Axioms plus the Axiom of Choice (ZFC); Peano Axioms of Arithmetic; Gödel’s
Incompleteness Theorems; Being and Event Parts I, II, IV and V; Some Alternative Axiomatizations of
Mathematics and Set Theory

Session II (Tuesday, 31 May)

2-4pm – Cardinal and Ordinal Numbers
Cardinal Numbers; Well-Orderings; Ordinal Numbers; Continuum Hypothesis; Being and Event Part III
and Meditation 26; Introduction to Surreal Numbers

6-8pm – Kurt Gödel and the Constructible Hierarchy
Some Relevant Model Theory; Formal Semantics of Situations; Formal Semantics of Events; Gödel’s
Completeness Theorem; Compactness Theorem; Löwenheim-Skolem Theorem and Paradox; Transfinite
Induction; Cumulative and Constructible Hierarchies; Axiom of Constructibility; Gödel’s Proof of
Consistency; Being and Event Meditation 29; Introduction to Large and Inaccessible Cardinals

Session III (Tuesday, 7 June)

2-4pm – Paul Cohen, Forcing and Generic Models
General Machinery of Forcing; Analogy between Forcing and Field Extensions; Cohen’s Proof of
Independence

6-8pm – Beyond Cohen’s Forcing
Cohen’s Proof of Independence (con’t); Being and Event Meditations 33, 34 & 36; Forcing in terms of
Kripkean Semantics; Forcing in terms of Boolean-valued models; Forcing Axioms and Generic
Absoluteness; Introduction to Categorial Logic and Topos Theory; Forcing in terms of Topos Theory;
Short Introduction to Forcing in Badiou’s The Logics of Worlds

Suggested Introductory Reading List

Avigad, J. (2004). “Forcing in Proof Theory”. http://www.andrew.cmu.edu/user/avigad/Papers/forcing.pdf.
Badiou, A. (2007). Being and Event. Translated by Oliver Feltham. London: Continuum.
Badiou, A. (2007). The Concept of Model. Translated by Zachary Fraser & Tzuchien Tho. Victoria: re:press.
Badiou, A. (2008). Number and Numbers. Translated by Robin Mackay: Cambridge: Polity.
Badiou, A. (2009). The Logics of Worlds. Translated by Alberto Toscano. London: Continuum.
Bowden, S. (2005). “Alain Badiou: From Ontology to Politics and Back”.

http://jffp.pitt.edu/ojs/index.php/jffp/article/view/244/238.

Chow, T. (2004). “Forcing for Dummies”. http://math.mit.edu/~tchow/mathstuff/forcingdum.
Chow, T. (2008). “A Beginner’s Guide to Forcing”. http://arxiv.org/abs/0712.1320.
Cohen, P. (2002). “The Discovery of Forcing”.

http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.rmjm/1181070

010.
Cohen, P. (2008). Set Theory and the Continuum Hypothesis. New York: Dover.
Crossley, J., et. al. (1991). What is Mathematical Logic?. Mineola: Dover.
Devlin, K. (1991). The Joy of Sets: Fundamentals of Contemporary Set Theory. New York: Springer.
Doxiadis, A., et. al. (2009). Logicomix: An Epic Search for Truth. Bloomsbury Publishing PLC.
Easwaran, K. (2007). ” A Cheerful Introduction to Forcing and the Continuum Hypothesis”.

http://arxiv.org/abs/0712.2279.

Fraser, Z. (2006). “The Law of the Subject: Alain Badiou, Luitzen Brouwer and the Kripkean Analyses of
Forcing and the Heyting Calculus”. http://www.cosmosandhistory.org/index.php/journal/article/view/30.
Goldblatt, R. (2006). Topoi: The Categorial Analysis of Logic. Mineola: Dover.
Hallward, P. (2003). Badiou: A Subject to Truth. Minneapolis: U. of Minnesota Press.
Halmos, P. (1974). Naive Set Theory. New York: Springer.
Jech, T. (2004). Set Theory: The Third Millennium Edition, Revised and Expanded. New York: Springer-Verlag.
Jech, T. (2008). “What is Forcing?”. http://www.ams.org/notices/200806/tx080600692p.pdf.
Kanamori, A. “Cohen and Set Theory”. Bull. Symbolic Logic 13(3) (2008), 351-78.
Kunen, K. (1992). Set Theory: An Introduction to Independence Proofs. New York: North Holland.
Norris, C. (2009). Badiou’s Being and Event. London: Continuum.
Pluth, E. (2010). Badiou: A Philosophy of the New. Cambridge: Polity Press.
Potter, M. (2004). Set Theory and its Philosophy. Oxford: Oxford U. Press.
Smullyan, R. & M. Fitting. (2010). Set Theory and the Continuum Problem. Mineola: Dover.
Tiles, M. (1989). The Philosophy of Set Theory. Mineola: Dover.

The London Consortium is a multi-disciplinary graduate programme in Humanities and Cultural Studies. We are a collaboration between five of London’s most dynamic cultural and educational institutions: the Architectural Association, Birkbeck College (University of London), the Institute of Contemporary Arts, the Science Museum, and Tate.

SUTURE ENTRY

Would you look at that? It looks like I forgot an entry. [[UPDATE: It seems I hadn't forgotten it at all---it was published on this blog back on March 5th. Somehow, due to my clumsy bloghandling, it disappeared, probably while I was trying to do some editing to it. As my wife put it, the suture unravelled somehow...]]

New and original material coming soon, by the way. Probably something like an overview of a series of seminars I'm going to be giving in Mexico City next month (on May 10th and 11th -- details to come!) on Badiou's Concept of Model and use of the forcing concept, before launching into desperate attempt to break out of Badiou's work with some reflection on Jean-Yves Girard's research programme in mathematical logic, the subject of my current research.

For now, though, I bring you SUTURE. Again.

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BADIOU DICTIONARY
SUTURE entry

The word ‘suture’ takes on three distinct meanings in Badiou’s texts. These do not mark distinct periods in the evolution of a single category so much as three different categories whose association under the same name perhaps signals nothing more interesting than synonymy—though some hesitation in accepting this conclusion is no doubt appropriate. To keep things as clear as possible, we will label these categories ideological suture, ontological suture, and philosophical suture, and we will deal with them in turn.

Ideological Suture

The word ‘suture’, in a sense which Badiou will diagnose as exclusively ideological in scope, first appears in Badiou’s work in late 1960s. It was the subject of an intense debate amongst the members of Le Cercle d’épistémologie (the working group behind Les Cahiers pour l’Analyse), which was polarized by the positions of Jacques-Alain Miller on one side, and Alain Badiou on the other. The first move was Miller’s. His contribution to the first issue Les Cahiers pour l’Analyse, ‘Suture (elements for a logic of the signifier)’, he sought to extract the concept of suture from the implicit state it enjoyed in Lacan’s teachings. By Miller’s reading, Lacan had recourse to the word ‘suture’ on a handful of occasions to name the covering of an essential lack in discourse, by way of an short-circuiting of heterogeneous orders (the imaginary and the symbolic, for instance), an operation that serves to constitute the subject by installing it in a chain of signifiers. Miller’s gambit, above and beyond his effort at exegesis, is to show that the operation of suture is at work even in those discourses where we expect it least, claiming to detect it in Gottlob Frege’s rigorously anti-psychologistic attempt to derive the laws of arithmetic from the foundations of pure logic. The focus of the article is Frege’s definition of zero as ‘the Number which belongs to the concept “not identical with itself”’ which, according to Frege’s earlier definition of a Number as a set of concepts whose extensions are equal, comes down to defining zero as the set of concepts F whose objects can be put in a one-to-one correspondence with the objects describable as ‘not identical to themselves’. On Miller’s reading, it is the exigency to preserve ‘the field of truth’ in which arithmetic must be inscribed that forces Frege to consider the extension of the concept ‘not identical to itself’ to be empty—for this field would suffer ‘absolute subversion’ if a term, being non-self-identical, could not be substituted for itself it the signifying chain (Miller 28-9). (This slippage from an object to the mark that indicates it goes unnoticed by Miller—a thread that Badiou will later seize upon in both MM and NN.) This definition, Miller claims, ‘summons and rejects’ (Miller 32) the non-self-identical subject, whose unconscious effects can be detected in the Fregean operation of succession (the ‘plus one’) that takes us from one number to another. That operation, Miller argues, functions only insofar as it is possible for the non-identical (the subject), lacking from the field of truth, to be ‘noted 0 and counted for 1’ (Miller 31). He grounds this argument on Frege’s definitions of one and the successor: Frege defines one as ‘the number of the extension of the concept: identical with zero’ (Frege §77, p.90), and defines the successor of n as the Number of the concept ‘member of the series of natural numbers ending with n’ (Frege §79, p.92)—a definition which could only yield n itself the zero which belongs to each of these series were not, again and again, counted as one. But this ‘counting of zero for one’, by Miller’s lights, depends entirely on the suturing of the subject that engenders the field of logical truth. It is therefore the subject that makes succession tick—but a subject manifested only in the suturation of its lack and so a condemned to miscognition on logic’s behalf.

Badiou will have none of this. The counterargument he delivers in ‘Mark and Lack: On Zero’ (which appears in the tenth and final volume of the Cahiers) can be condensed as follows:

(1) Scientific discourse in general, and mathematical logic in particular, is not a unitary field of discourse or ‘field of truth’ at all. It must be conceived, instead, in terms of multiple stratified apparatuses of inscription.

(2) At no point does any discursive operation in any of these strata have any occasion or need to invoke a radical, unthinkable ‘outside’. What looks like an invocation of ‘lack’—the statement that the concept ‘not-identical-to-itself’, for example, has an empty extension—is nothing but a referral to an anterior stratum of the discourse. No scientific inscription enjoys the paradoxical status of ‘cancelling itself out’, as Miller supposed to take place in the Fregean invocation of the ‘non-self-identical’. Analysing Frege’s definition of zero, for instance, we should see the inscription, on a particular stratum (which Badiou terms the ‘mechanism of concatenation’, or ‘M2’, and whose task is merely to assemble grammatical expressions), of the predicate ‘x is not identical to x’ as a perfectly stable inscription (which indeed presupposes the self-identity of the mark ‘x’ in a perfectly consistent fashion and without the slightest ambiguity). It is only on another stratum (M3, the ‘mechanism of derivation’, which sorts the output of M2 into theorems and non-theorems) that the system ‘rejects’ the existential quantification of ‘x is not identical to x’ as a non-theorem. In no sense does M3 cancel out the productions of M2, or summon them only to reject them: it receives these productions as its raw material, and operates on them in a fashion altogether different what we find in M2. On a subsequent stratum (M4), the predicate 0 can then be defined in terms of the predicate whose extension was shown to be empty, and so on. What transpires in all of this is not, and cannot be, the ephemeral invocation of the non-self-identical subject, or a wound in discourse obscured by the scar of the letter, but a stable relay between fully positive strata the assemblage of which ‘lacks nothing it does not produce elsewhere’ (MM 151), a rule which, Badiou affirms, holds good for all of science.

(3) Not only does the stratification of the scientific signifier exclude suture from science, it suffices to foreclose the subject from scientific discourse altogether, and this is the secret of science’s universality: Science is a ‘psychosis of no subject, and hence of all: congenitally universal, shared delirium, one has only to maintain oneself within it in order to be no-one, anonymously dispersed in the hierarchy of orders’ (MM 161).

(4) Rigorous stratification and foreclosure of subjective suturation are not just accidental features of science, but what constitute science as science. It is they that give form to the notion of the epistemological break, the continuous struggle by which science separates itself from ideology.

(5) ‘The concept of suture,’ therefore, ‘is not a concept of the signifier in general, but rather the characteristic property of the signifying order wherein the subject comes to be barred – namely, ideology’ (MM 162).

This is not to say that suturation never happens when scientists speak. It occurs repeatedly – but these occurrences are nevertheless extrinsic to science in itself. The suturing of scientific discourse is what occurs in the continual establishment of epistemological obstacles, the destruction of which is the sciences’ incessant task. This dialectic of stratification and suturation, or of science and ideology, is elaborated in the appendix to ‘Mark and Lack,’ in a detailed case study of Gödel’s first incompleteness theorem—a study which implicitly attacks Lacan’s attempt to exploit this theorem in ‘Science and Truth’. (See entry on ideology for more details.)

Ontological Suture

Upon mathematics’ ontological baptism, at the beginning of Being and Event, the word ‘suture’ makes a prominent return. It comes to serve two functions: to name the empty umbilicus that links each situation to being by way of the void that haunts it (‘I term void of a situation this suture to being’ (BE 55)), and to christen being with the ‘proper name’ Ø, the mathematical sign of the empty set. Given the rigour and severity of his attack on Miller’s application of the notion of suture to mathematical discourse, Badiou’s abrupt decision to declare Ø set theory’s ‘suture-to-being’ (BE 66) – in a sense ‘which will always remain enigmatic’ (BE 59) – may strike the reader as surprising. More surprising still is that no link, positive or negative, is drawn between the then-falsified Millerian thesis that the subject’s inconsistency is sutured by the arithmetical 0, and the now-affirmed thesis that being’s inconsistency is sutured by the set-theoretical Ø. Even in Number and Numbers, where Miller’s thesis comes in for a second round of attacks, we find the new metaontological suture-thesis affirmed with innocence throughout the book (see NN Chapter 3).

When pressed on this point, Badiou responds that between these two theses, the word ‘suture’ ‘changes its meaning’: it is no longer a question of invoking the void of the (Lacanian) subject, but the void of being as radical inconsistency. This was not in doubt. But the argument deployed in ‘Mark and Lack’ against applicability of the notion of suture to the Fregean 0 nowhere depends on the identification of lack, or radical inconsistency, with the subject. If the argument is sound then it will remain so under the uniform substitution of ‘being’ for ‘subject’, and one cannot use this substitution to flee the difficulties encountered by Miller: we cannot avoid seeing that ‘the torch which lights the abyss, and seals it up, is itself an abyss.’ If the meaning of ‘suture’ in Being and Event differs from the meaning of ‘suture’ in ‘Mark and Lack’ only with respect to the terms it relates—subject then, being now—then the Badiou of 1988 and after remains hostage to the Badiou of ’69, and the stratified psychosis of mathematics will absolve itself from ontology as relentlessly as it does from ideology, foreclosing being as radically as it does subjectivity.

Philosophical Suture

There is a third sense in which Badiou uses the word ‘suture’, which is not so obscurely entangled nor obviously connected with its older usage, though certain structural similarities can still be observed: here, it names a particular – potentially disastrous – way in which philosophy may relate itself to one of its conditions. The relation of conditioning that the philosopher is charged with maintaining between extra-philosophical disciplines (truth procedures) and her own collapses into a relation of suture when, by way of destratification, the philosopher confuses these two disciplines with one another. It is helpful to make a distinction here, according to which partner in the suture achieves dominance. The dominance of the condition—such as the poetic condition dominates the late Heidegger and his pupil, Gadamer, the political condition dominates certain strains of Marxist thought, the scientific condition dominates Carnap and Hempel, and the amorous condition dominates Levinas and Irigaray—is indicated by its hegemony over the philosophical category of truth and its capture of philosophical rationality. No other modes of truth but those of the condition, sutured in dominance, are recognized, and the philosopher measures her reasoning by strictures proper to the conditioning discipline. This renders philosophy incapable of fulfilling its mandate, which is to construct a systematic compossibilization of heterogeneous truths. The dominant position in a suture may also be occupied by philosophy. When this occurs, philosophy takes itself as producer of truths – the kind of truths, moreover, that ought to be entrusted to an external condition. When this takes place, the threat of disaster looms large, and so I refer the reader to the entry on that concept.