Badiouandapos;s Being And Event And The Mathematics Of Set Theory

Alain Badiou's __Being and Event__ continues to impact philosophical investigations into the question of Being. By exploring the central role set theory plays in this influential work, Burhanuddin Baki presents the first extended study of Badiou's use of mathematics in __Being and Event__. Adopting a clear, straightforward approach, Baki gathers together and explains the technical details of the relevant high-level mathematics in Being and Event. He examines Badiou's philosophical framework in close detail, showing exactly how it is 'conditioned' by the technical mathematics. Clarifying the relevant details of Badiou's mathematics, Baki looks at the four core topics Badiou employs from set theory: the formal axiomatic system of ZFC; cardinal and ordinal numbers; Kurt Gödel's concept of constructability; and Cohen's technique of forcing. Baki then rebuilds Badiou's philosophical meditations in relation to their conditioning by the mathematics, paying particular attention to Cohen's forcing, which informs Badiou's analysis of the event. Providing valuable insights into Badiou's philosophy of mathematics, __Badiou's Being and Event and the Mathematics of Set Theory__ offers an excellent commentary and a new reading of Badiou's most complex and important work. Cover Half-title Title Copyright Contents List of Figures and Tables Acknowledgements Note on Abbreviations, Citations and Translations Introduction The question of the event Statement of purpose and delimitation of purview 1 Mathematics = Ontology Mathematics, ontology and philosophy The other conditions of philosophy and the compossibilization of truths The grand style versus the little style of philosophical inquiry Badiou is not a structuralist Being is not essentially mathematical Metaontology versus the mathematical sciences Metaontology versus humanistic philosophy 2 Ontology of Axiomatic Set Theory Mathematical concept of the set Intensional specification of multiples Russell’s Paradox and the Axiom of Separation Relation of inclusion and the notions of subset and power set Basic set operations Supplementary properties, relations and functions Notion of the formal axiomatic system (FAS) Introduction to model theory Deduction, consistency, completeness and undecidability Gödel’s Completeness Theorem and the two types of consistency The Zermelo-Fraenkel Axioms of Set Theory plus the Axiom of Choice Gödel’s Incompleteness Theorems and the foundation of mathematics 3 Metaontology of Situations and Presentation Precedents to the concept of situation Two confusions about situations Situation as set Situation as model The flat plane of presentation Universes and quasi-complete situations The ontological decision that the one is not Consistency and militant commitment Following through the multiplicity of being ZFC as the a priori conditions for ontology Resolving the two Hurdles ZFC as a laicized and consistent science of inconsistent multiplicity Metaontology of the Axiom of the Void Axiom of Existence Foundational edge-of-the-void elements Inconsistency and the void The void as the ontological atom 4 Metaontology of the State and Representation Properties, subsets and representations Names and singletons The power set and the regime of representation Power set of Cartesian products and the regimes of relation Representation versus predication The state prevents the situation from encountering its own inconsistency The state of the empty set and of a quasi-complete situation 5 Ontology and Metaontology of the Cardinal and Ordinal Numbers Basic extensions to the notion of number Cardinal numbers, set sizes and one-to-one correspondences Cantor’s Theorem and the uncountability of the continuum The Continuum Hypothesis The first few ordinal numbers Ordinal numbers and well-orderings An ordinal is the set of the ordinals preceding it Ordinals and homogeneous transitivity Cardinals as specific ordinals Summary of the mathematics Ontology versus onticology of nature The typology of relations between structure and metastructure Nature and homogeneous transitivity The foundational element of historical situations The ontological stability and homogeneity of nature 6 Ontology and Metaontology of the Constructible The Löwenheim-Skolem Theorem and Skolem’s Paradox Introduction to Gödel’s result on constructibility Transfinite induction and defining the sequence of ordinals Defining the ordinals via transfinite induction The cumulative hierarchy of pure sets V The constructible hierarchy of pure sets L Proving the Axiom of Choice Proving the Continuum Hypothesis Summary of the Mathematics of the Constructible Metaontological orientations of thought The constructivist orientation of thought Nominalist metaontology Knowledge and encyclopaedic determinants 7 Ontology of Forcing and Generic Sets The specific employment of forcing in Being and Event The semantic versus syntactic approach to establishing consistency Universe-building The general technology of forcing Reducing the ground model to a countable transitive model of ZFC The poset P of forcing conditions Concept of a filter G is a generic filter The existence of a generic filter and its exteriority with respect to the ground model The generic filter from the viewpoint of the ground model The forcing language and the P-names The values of the P-names and the construction of the generic extension M[G] The relation of forcing and the Forcing Theorems The Generic Model Theorem Two ad hoc features in Cohen’s proof Finite partial functions 8 Metaontology of the Subject, Truth, the Event and Intervention Forcing from within M and S Forcing from within M and S (summary) Initial hints of Badiou’s metaontology of forcing Badiou’s schema of metaontological translation Pure versus empirical philosophy of ontology The ‘trigger’ and the ‘dynamic’ Correct subsets, knowledge and the encyclopaedia The generic set and knowledge Supporting the trajectory of constructing Cantor’s technique of diagonalization Aleatory diagonalization by the subject The subject versus the state The being of truth, the subject-languageand the generic extension S(♀) The evental site and the matheme of the event Non-ontology of the event Undecidability of the event Nomination and recognition of the event as event Unconstructibility of the event The legalization of intervention via choice Following through with intervention Epilogue Notes References Index