Notes on “the digital”

It is mathematically demonstrable that the ontology of set theory and the ontology of the digital are not equivalent.

The realm of the digital is that of the denumerable: to every finite stream of digits corresponds a single natural number, a finite ordinal. If we set an upper bound on the length of a stream of digits – let’s say, it has to fit into the available physical universe, using the most physically compact encoding available – then we can imagine a “library of Boole”, finite but Vast, that would encompass the entirety of what can be digitally inscribed and processed. Even larger than the library of Boole is the “digital universe” of sequences of digits, D, which is what we get if we don’t impose this upper bound. Although infinite, D is a single set, and is isomorphic to the set of natural numbers, N. It contains all possible digital encodings of data and all possible digital encodings of programs which can operate on this data (although a digital sequence is not intrinsically either program or data).

The von Neumann universe of sets, V, is generated out of a fundamental operation – taking the powerset – applied recursively to the empty set. It has its genesis in the operation which takes 0, or the empty set {}, to 1, or the singleton set containing the empty set, {{}}, but what flourishes out of this genesis cannot in general be reduced to the universe of sequences of 0s and 1s. The von Neumann universe of sets is not coextensive with D but immeasurably exceeds it, containing sets that cannot be named or generated by any digital procedure whatsoever. V is in fact too large to be a set, being rather a proper class of sets.

Suppose we restrict ourselves to the “constructible universe” of sets, L, in which each level of the hierarchy is restricted so that it contains only those sets which are specifiable using the resources of the hierarchy below it. The axiom of constructibility proposes that V=L – that no set exists which is not nameable. This makes for a less extravagantly huge universe; but L is still a proper class. D appears within L as a single set among an immeasurable (if comparatively well-behaved) proliferation of sets.

A set-theoretic ontology such as Badiou’s, which effectively takes the von Neumann universe as its playground, is thus not a digital ontology. Badiou is a “maximalist” when it comes to mathematical ontology: he’s comfortable with the existence of non-constructible sets (hence, he does not accept the “axiom of constructibility”, which proposes that V=L), and the limitations of physical or theoretical computability are without interest for him. Indeed, it has been Badiou’s argument (in Number and Numbers) that the digital or numerical enframing of society and culture can only be thought from the perspective of a mathematical ontology capacious enough to think “Number” over and above the domain of “numbers”. This is precisely the opposite approach to that which seeks refuge from the swarming immensity of mathematical figures in the impenetrable, indivisible density of the analog.

2 thoughts on “Notes on “the digital””

  1. worth adding that we could lift the restriction on digit-streams being finite. that way we get a set C which is the size of the continuum (the standard decimal representation of a real number is as an infinite digit stream). this is strictly bigger than D but still tiny in comparison to V.

    one caveat tho: Badiou (rightly in my view) is suspicious of notions like “too big to be a set” which rely far too heavily on ideological common sense. he talks of V being *inconsistent* as a multiple, it cannot “hang together” so to speak.

  2. This is a potential difficulty for those who identify facts with claims. The number of expressible claims must be vast, but is presumably denumerable.

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