Number and numbers
The political regime of global capitalism reduces the world to an endless network of numbers within numbers, but how many of us really understand what numbers are? Without such an understanding, how can we challenge the regime of number?
In Number and Numbers Alain Badiou offers an philosophically penetrating account with a powerful political subtext of the attempts that have been made over the last century to define the special status of number. Badiou argues that number cannot be defined by the multiform calculative uses to which numbers are put, nor is it exhausted by the various species described by number theory. Drawing on the mathematical theory of surreal numbers, he develops a unified theory of Number as a particular form of being, an infinite expanse to which our access remains limited. This understanding of Number as being harbours important philosophical truths about the structure of the world in which we live.
In Badiou's view, only by rigorously thinking through Number can philosophy offer us some hope of breaking through the dense and apparently impenetrable capitalist fabric of numerical relations. For this will finally allow us to point to that which cannot be numbered: the possibility of an event that would deliver us from our unthinking subordination of number.
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Von Neumann Ordinals
Succession and Limit The Infinite
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algebraic axiom Axiom of Foundation Badiou belonging biunivocal correspondence Cantor concept of number Dedekind defined denote dense order discriminant domain of Numbers dyadic rationals empty set example exists fact follows form of N2 Frege function given Gonshor high set immanent inconsistent induction infinite system infinity larger limit ordinal logic low set mathematical matter of N2 maximal minimal element minimal matter multiple-being natural multiples natural whole numbers negative Number Number of finite object ontological operational order of Numbers order-relation ordinal level ordinal-matter ordinals smaller Peano philosophical positive Number precisely pure multiple rational numbers real numbers relation residue rule Russell's paradox sense set of ordinals set theory sets of Numbers singleton situation smallest ordinal structure sub-Number successor ordinal suppose surreal numbers symmetric counterpart term theorem thinking of number tion total order trans transfinite induction transitive set truth upper bound void well-orderedness zero