Κυριακή, 12 Αυγούστου 2012

Η επανάσταση του Zermelo και ο Badiou

Σε κείμενο του Lyn Sebastian Purcell για την σχέση Badiou Derrida γίνεται εκτενής ανάλυση της συνολοθεωρίας.Παραθέτω ένα απόσπασμα και σχετικό σύνδεσμο .


ZERMELO'S REVOLUTION We should like to begin our engagement with Badiou by noting a ghostly presence within Badiou’s own thought—a specter (revenant) who haunts the whole of his ontology. Consider the following statement from Being and Event: ‘That it is necessary to tolerate the almost complete arbitrariness of a choice, that quantity, the very paradigm of objectivity, leads to pure subjectivity; such is what I would willingly call the Cantor-Gödel-Cohen-Easton symptom’ (BE 280). We are not here interested in this full itinerary, which is punctuated by the names of four great mathematicians, but only its first point, and the unmentioned name that stands between Cantor and Gödel, namely Ernst Zermelo. This mathematician, who is present only as a dash in Badiou’s thought, we argue forms the symptomal point of his enterprise. If attended to correctly, we argue it is here that one can uncover an alternative appropriation of Cantor. 2.1 Against the Whole The ‘Cantorian Revolution’ in Badiou’s thought is tantamount to the rejection of the whole. After Cantor established that it was possible to think the infinite, reversing more than two millennia’s wisdom on the matter, there was a short period in which set theory operated by use of something like Gottlob Frege’s unlimited abstraction principle, which had the advantage of allowing mathematicians to obtain almost all the sets necessary for mathematics from it alone.3 It was as follows: given a well defined property P, there exists a unique set A that consists of only those things that have the property P. Usually, such a set is expressed with braces as follows: {x | P(x)}, which means ‘the set of all x having the property x’. The difficulties with this principle are well-known: such a principle allows for selfmembership. If some sets can be members of themselves, then others are sets that are not members of themselves. That this distinction results in a logical paradox was an observation Bertrand Russell made (and Zermelo independently), and has come to be known as Russell’s paradox.4 The response of the mathematical community was to try to avoid this inconsistency by addressing or reformulating the abstraction principle. This aim was the point of Russell’s theory of types. Yet, in the end the solution that was provided by Ernst Zermelo (in 1908) proved most acceptable. 3We note that Badiou rightly counts Frege as the second attempt to think a set, while Cantor’s intuition of objects constitutes the first (BE 40). 4 For those interested, Badiou reproduces this paradox in Being and Event pages 40-1, and more thoroughly in Logics of Worlds, trans. Oliver Feltham, New York, Continuum Press, pp. 153-5 (Henceforth: LW

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