Badiou and The Continuum Hypothesis

(Previously) ...Next, the mea culpa. It concerns Badiou's position on the Continuum Hypothesis. In 1878 Georg Cantor offered a hypothesis about the continuum, essentially a hypothesis about the nature of continuous number. More specifically the hypothesis concerned Cantor's famous two sizes of infinity and the relation between them. After establishing the position of the smaller infinity, the infinity of rational or arithmetical number, Cantor hypothesized that the "next highest" number would be his larger infinity, the infinity of real or geometric number. I write next highest in scare quotes because the question of size and even counting itself stops making intuitive sense after transgressing the threshold of finitude.

If you're looking for a more technical statement of the CH, here is a helpful one by Mary Tiles from her excellent book The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise (p. 1):

"20 = ℵ1, which has come to be known as Cantor's continuum hypothesis, thus says that the number of points on a line is the second infinite cardinal number: there are none in between ℵ0 and 20."

Okay, so the CH has a long history in mathematics and I'll be the first to admit that my knowledge of the underlying math is wobbly at best. But let's establish the basics at the outset.

First, Cantor posed the hypothesis, and we know that he thought it was true, yet struggled unsuccessfully to find a proof during his lifetime. Yet later in 1940, Kurt Gödel used the tools of set theory to demonstrate the truth of the Continuum Hypothesis (crowd cheers). A few years later in 1963 Paul Cohen made a model of set theory where the CH failed (crowd boos). So Gödel showed the CH was true, but Cohen showed it was false -- who's right? When a hypothesis lands in this predicament, seemingly both true and false, mathematicians label the hypothesis "independent." Independent means that the hypothesis is simply not within the realm of relative truth or falsity, given whatever foundation was originally marshaled, in this case ZFC set theory. It simply doesn't make sense to say that the CH is true or false; all we can say is that it's "independent." So that's where we are today. The hypothesis is still out there, floating around, as yet unprovable (nor deniable) given current tools and methods. Although there is nothing that will stop the CH from being demonstrated (or denied) at some point in the future, if new methods are discovered.

Now we get to the mea culpa. On the Being & Event podcast, I made a comment something to the effect of "deep down, Badiou must think the Continuum Hypothesis is true." Perhaps this was a projection on my part, given that, for philosophical reasons rather than mathematical ones, I also consider the CH to be a valid description of discrete and continuous rationality. And given my reading of Badiou in Being and Event, I prognosticated that Badiou would support the validity of the CH on similar grounds as well.

We recorded most of the podcast already a year or more ago. At that time I had skimmed Badiou's newest large treatise, The Immanence of Truths, but only recently started reading the book more systematically. Imagine my surprise when I got to page 67 (in the English edition), where Badiou states in no uncertain terms that he suspects the Continuum Hypothesis is false:

"New axioms of infinity and new theoretical conjectures may soon force the falsity of the continuum hypothesis" (67);

"I have thought...that [the hypothesis] is quite simply false" (67).

My guess had been that Badiou would say yes, but now he says no. Oops. So here are my thoughts on this apparent discrepancy. First, as stated above, if we take set theory as a basis, CH is neither true nor false as of today, but "independent." So any speculations (including my own) about it being false or true should be taken extremely loosely.

Second, I'm not even sure Badiou is clear on his own stance. Even in these pages of The Immanence of Truths he admits being seduced by Hugh Woodin -- a current luminary in the field of transfinite mathematics -- then suggests an eventual return to Cantor, but we know Cantor suspected that CH was true. And then later in IoT Badiou says, with much bravado, "I choose Cohen" (which was, in a sense, his culminating mantra from Being and Event). So it's unclear if Badiou is aiming forward, or plotting a return.

Third, it might also depend on how we interpret the Continuum Hypothesis itself. (It *needs* an interpretation, in my view, not simply a proof. A bit like how we have competing interpretations of quantum mechanics.) And for me the essence of CH is the specific gap between natural infinity and real infinity. Or in my parlance this would be the specific gap between digital mediation and analog mediation. This is essentially why, on the podcast, I claimed that the Badiou of 1988, the Badiou of Being and Event, would likely endorse the truth of the hypothesis. To reiterate, Being and Event entirely hinges on the abyss between the rational and the real (out of which emerges the event/truth/subject/etc). In other words the structure of the Continuum Hypothesis can be superimposed, like a map, on the very narrative of Being and Event as a whole.


Or at least the hypothesis unlocks the book in profound ways. As suggested already above, this is demonstrated most vividly in Mediations 26 & 27 (particularly at p. 278 of the English edition), which in my view is the most import part of the treatise. Badiou's story begins in the domain of the smaller infinity, the infinity of discrete, ordered number. It ends at the larger infinity, the "indiscernible" infinity of generic multiplicity. And there's a gap in between. Aleph-zero, aleph-one, with a gap between -- that's what I mean when I say that the Continuum Hypothesis provides a fitting map for the argument of Being and Event.

Interestingly The Immanence of Truths argues for something very similar (Badiou's "quite simply false" quote notwithstanding), namely that the finite is the result of an intersection between two infinities of different types. That screams CH-truther to me! In other words, while Badiou clearly says that he expects the CH to fall, there's a performative contradiction between what he says and the specific arguments of both Being and Event and Immanence of Truths.

Last thought about the Continuum Hypothesis... I think the Woodins of the world are basically "Cantor's Paradise" people, if I can call them that. In other words, these thinkers tend to emphasize the *multiplicity* of infinities. That's all well and good for research into large cardinals. But CH isn't about that; it's not about the paradise. The hypothesis isn't about the multiplicity of infinities, but rather about two specific ones, natural infinity and real infinity. In other words it's about a *duality* of infinities rather than a multiplicity. And, if we're lucky, this duality can be restructured not so much as a duality but as a proper dialectic. (But that's a story for another day!) In sum, I consider Badiou to be fundamentally seduced by both of these tendencies. Badiou is a dualist/dialectician, but he's also a philosopher of the multiple. And it's not always clear which side wins out.

In other words, while Badiou has clearly made statements to the contrary, I intend to stand firm in my conviction. The Continuum Hypothesis is an valid description of natural and real rationality. And I suspect, deep down, Badiou believes this too, if not in word then certainly in deed.