Forcing in Being and Event

Πηγή: Form & Formalism

 

Forcing in Being and Event

Between the theory of forcing presented in ‘Infinitesimal Subversion’ and the one we find in Being and Event, intervenes the a new and decisive condition: a technique developed by Paul J. Cohen in his proof of the independence of the Generalized Continuum Hypothesis and the Axiom of Choice from the axioms of Zermelo-Fraenkel set theory (ZF), which likewise appears under the name of ‘forcing’. Before we address its incorporation into Badiou’s philosophical apparatus, we will take a quick look at forcing in its native, mathematical terrain.

It is, once again, set-theoretical model theory that provides Badiou with the requisite conceptual (scientific) material. Like Robinson’s procedure for the making of ‘non-standard’ models, Cohen’s forcing technique is, at bottom, a systematic way of generating a new model from a model already given. The main thrust of Cohen’s proof is to take a countable, transitive model[4] of ZF and ‘force’ the existence of a new model by supplementing it with a generic element included in, but not belonging to, to initial model—together with all the sets which can be constructed on the supplement’s basis by licensed by the ZF axiomatic. Considered in its logical structure, forcing is a relation of the form ‘a forces P’, where a is a set and P a proposition that will hold in the generic extension of the initial model—provided that a turns out to belong to the generic supplement on which that extension is based. In this respect, forcing resembles a logical inference relation, but one that differs markedly from the inference relation of classical logic—the law of the excluded middle, in particular, does not hold for the forcing relation, and the logic it generates is essentially intuitionistic.[5]

As Cohen has shown, the consequences this supplementation can be quite extraordinary, and go far beyond simply adding a new set’s name to the census. The generic supplement, for instance, may be structured so as to induce a one-to-one correspondence between transfinite ordinals that, in the initial model/situation, counted as distinct orders of infinity, thereby collapsing them onto one another and making them effectively equal. Cohen exploited this possibility to great effect by taking the model that Gödel had built in order to show that the Generalized Continuum Hypothesis (GCH)—the thesis that the size of the set of subsets set of any transfinite cardinal number À_n is equal in size to the next greatest cardinal À_n+1—is consistent with ZF (a model in which the continuum hypothesis holds), and on its basis forcing a generic extension in which the continuum hypothesis fails (the extension being a model in which the set of subsets of À_n is demonstrably equal to almost any cardinal whatsoever, so long as it’s larger than À_n), thereby demonstrating the consistency of GCH’s negation with the theory, and hence the independence, or undecidability, of GCH with respect to ZF.

Being and Event recovers Cohen’s concept and enlists it in a re-articulation of the existing category of forcing: the set underlying the model is now seized upon as the situation that forcing will transform, and faithful Miller’s cartography of change, Badiou adds that the whole procedure—both the articulation of the generic truth and the forcing of its consequences for the situation to come—must in every case proceed from an anomalous occurrence in the ‘utopic point’ of the situation in question, now rechristened ‘evental site’. Though it is now Cohen rather than Robinson whose mathematics condition Badiou’s theory of change, the new category of forcing preserves most of the features familiar to us from “Infinitesimal Subversion.” One crucial difference, however, is that the whole process is now seized as a logic of subjective action: Forcing is now names “the law of the subject” [CITE], the form by which a subject faithful to an event transforms her situation into one to which a still-unknown[6] truth (understood as a generic subset of the initial situation) well and truly belongs, by deriving consequences that the inscription of this new constant will have brought about.

In light of Being and Event’s decision to interpret ZF as the theory of being qua being, and forcing as the form of a subject’s truth-bearing practice, Badiou extracts two lessons from Cohen’s proof of the undecidability of GCH: first, that it demonstrates the existence of a radical ontological gap or ‘impasse’ between infinite multiplicities and the sets of their subsets (to which Badiou associated the notions of ‘representation’ or ‘state of a situation’), the exact measure of which is indeterminate at the level of being-in-itself; second, that this ontological undecidability is nevertheless decidable in practice, but only through the faithful effectuation of a truth, suspended from the anomalous occurrence of an event.