1. THERE IS NO WHOLE?
This essay defends an idea that is no longer fashionable: that there is a whole. There are many detractors of this notion, though recently Alain Badiou has provided a novel, non-ethical reason for its rejection: such a notion fails to make the turn into the Cantorian Revolution, and any position that fails to do so is doomed to repeat the failures of the metaphysics of presence. (1) In opposition to this consensus, and especially Badiou's criticism, what we hope to articulate here is a novel concept of the whole--one that withstands contemporary criticism. In order to do this, we are going to take up the work of Jacques Derrida, already panned by Badiou's supporters, in order to develop our new sense of the whole. (2) Specifically, we find in Derrida's early work on the experience of the undecidable a statement of what constitutes a defense of the whole, but which equally breaks with Hegel's conception of the absolute. Our thesis is the following: the experience of the undecidable is the experience of an inconsistent whole. Like Socrates at his trial, when after being found guilty suggests that he ought to be rewarded with free meals for life at the Prytaneum, the aim of this essay amounts to a defense of Derrida by an admission of guilt. Derrida was a thinker of 'finitude', he did not make the turn into the 'Cantorian Revolution', and he was a thinker of the 'whole'. For all these transgressions, we claim our just dessert: that Badiou himself may be wrong. Or perhaps even more strongly phrased: it is we who remain Cantorian, since Cantor after all did hold that there was a whole, while Badiou is rather the descendant of Ernst Zermelo.
The stakes of this encounter, then, should be sufficiently clear. The possibility of transcendental philosophy, a philosophy that would seek to ascertain the limits of thought, has been exhausted. This end, however, constitutes a new beginning. It is the birth of infinite thought. While Badiou has established one possible way through to this goal, if we are right, there are two such paths, and failing to meet the requirements of one need not be counted as failing to make the turn into infinite thought. A second closely related consequence of our thesis is the production of a program of research. The role of truth procedures, events, and the possibility of an inconsistent ontology are all suggested. Perhaps most promising, however, is that it provides the way to another model of subjective intervention, which we provisionally call beauty or nobility (kalos). This pay-off should entice us enough at least to entertain the possibility of another path into infinite thought.
2. 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-Godel-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 Godel, 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. …