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In 2006, Lucca Fraser published the article “The Law of the Subject: Alain Badiou, Luitzen Brouwer and the Kripkean Analysis of Forcing and the Heyting Calculus.”1 Still one of the best Anglophone commentaries on Badiou’s L’Être et l’Événement, she argued that Kripke forcing was sufficient to render the theory of the subject in Badiou’s L’Être et l’événement within an intuitionistic framework. This pluralization of logical frameworks allowed us some insight into the possibilities of mathematical ontology of which Badiou was not aware. That is, it forced us, and eventually Badiou, to recognize that his commitment to classical logic, where the principle of excluded middle was true, was unnecessary to the claims about the subject within his project. In turn, Badiou agreed and acknowledged this point but argued that this only applied to the subjective dimensions of mathematical ontology.2 Classical logic was still needed to express the core of the ontological analysis, based on Zermelo-Frankel set theory with the axiom of choice (hereafter ZFC).3

Sets, Set Sizes, and Infinity in Badiou's Being and Event