appendix
On the Development of Transfinite Set Theory

Set theory is one of the most significant conditions of Badiou's philosophy, and it provides the formal framework for his ontology. This appendix is designed to offer, independently of Badiou's unique application of the theory, some elementary background information concerning its history, purpose, and broad philosophical significance. The reader will find summary explanations of a number of concepts and terms referred to at various points during the main text, including actual and potential conceptions of the infinite, denumerable and nondenumerable sets, Cantor's continuum hypothesis, Russell's paradox, the Zermelo-Fraenkel axioms, Gödel's incompleteness theorem, and (in a little more detail) Cohen's generic sets and concept of forcing. All are presented with the strict minimum of technical complication and intimidation. 1

What the great German mathematician Georg Cantor (1845–1918) called “transfinite” or “suprafinite” set theory is clearly a theory that aims to combine a conception of the infinite with a conception of set (or number). 2 Its revolutionary innovation concerns the mathematical status of the “actual, ” or completed, infinite—the infinite conceived, precisely, as a set, that is, as an embraceable collection or whole. This appendix begins, then, with the general context in which this innovation took place. Subsequent sections look at the various components of the theory, and at perhaps the two most important moments (associated with Gödel and Cohen, respectively) of its

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