*What follows will no doubt butcher the history of mathematics. My aim here is to get right to the point of Alain Badiou’s intervention in set theory and ontology, shamelessly eliding and summarising important discoveries as I go and claiming no expertise. If anyone spots something technically incorrect about this post feel free to let me know. My education in mathematics ended at age 18 and we did not part amicably, and my enthusiasm is only now being rekindled through philosophy. I’m interested in understanding the logic, not in objections to the heuristic ‘value’ of set theory. *

After the gradual replacement of infinitesimals with the definition of limit, the mathematical and philosophical context in which Georg Cantor set to work was one that held the notion of infinity as irredeemably potential, essentially inaccessible, and more or less explicitly analogous to a transcendent entity, or God. It was also a time after the discovery of various non-Euclidean geometries and the dawning realisation that geometric axioms were not the foundation of physical extensive space but each relative to a given model thereof. If mathematical truths could no longer be grounded in representations of space, then perhaps they could be grounded in number, but we return to question of the coherence of number at its ultimate ‘limit’ of infinity.

Cantor began with a distinction between the enumeration of a given quantity of collected objects in a set on the one hand (ordinal numbers), and the relative size or ‘power’ of sets on the other (cardinal numbers). With finite sets we can say that the ordinal enumeration of its parts will be identical with its cardinal power; the finite whole is the sum of its collected parts. But when we consider infinite sets we realise this ratio breaks down and a one-to-one correspondence can be made between set and subset. In the infinite succession of natural numbers, then, there are as many odd numbers (subset) as there are odd and even numbers (set). We can, furthermore, pair off every natural number with a rational number, the latter infinite denumerable set adding no further cardinality to the former. All infinite denumerable sets have the same cardinality, regardless of how large a part of the ‘original’ infinite denumerable set they include. This ‘original’ set of all denumerable natural and rational numbers was denoted by Cantor as the Hebrew letter aleph, subscript zero: ℵ0

[I had the subscript formatted in my Word document but WordPress doesn’t have the toolbar options to hand on the editor. I have neither the time nor patience to fiddle around with the toolbar, so here the subscript characters will be in normal type.]

Things get interesting when Cantor considers the already-proven existence of irrational numbers. ℵ0 clearly does not exhaust the numerical description of a linear geometric continuum (all conceivable points on a line), hence the set-theoretic discovery of a distinct and nondenumerable order of infinity over and above, so to speak, the denumerable collection of rational numbers. This nondenumerable set Cantor called c [Fraktur script]. It follows that all conceivable segments of the geometric continuum (from ℵ0-dimensional to one-dimensional) are of the same cardinal power. Moreover, the axiom of the power set allows Cantor to denote the infinite arrangements of subsets found within set ℵ0 as 2^{ℵ0} proceeding on with a sequence of numbers each infinitely larger than the last. Cantor’s continuum hypothesis set out to prove that the power set axiom was the key to a well-ordered transfinite realm of successive cardinal numbers; that the nondenumerable infinite set of irrational numbers was actually the power set of ℵ0 all along, that 2^{ℵ0}=c, and that this set is the very next discrete cardinal number in an ordered sequence thereof, denoted by aleph, subscript one: ℵ1.

To Cantor’s great despair, he was unable to draw this direct link, and later Paul Cohen firmly established that there is no clear relationship between the ‘successor’ cardinal ℵ1 and the power set axiom. Cantor ultimately had to accept a distinction between the ‘consistent’ transfinite and the ‘inconsistent’ Absolute infinite inaccessible to numerical treatment, anticipating Russell’s paradox of the set of all sets (which is itself a set, etc.). To this day the precise boundary between consistent and inconsistent number remains unclear; the consequences of the power set axiom, the generation of ever further orders of cardinality, simply cannot be brought into alignment with the ordinal. The axiomatization of set theory, beginning with Ernst Zermelo, was an attempt to maintain transfinite order precisely by removing the inconsistent Absolute from the picture altogether. Definition of set would be replaced by axioms for avoiding inconsistency and paradox.

This is where many philosophers of mathematics leave the room. If axiomatic set theory cannot even define its own elementary concept, how can it be assigned any ontological authority? For Alain Badiou, however, this is precisely the place of ontology, *because* it does not authoritatively define its ground but instead modestly prescribes a set of procedures, marking a point where the subject can be brought into being through a decisive act. For Badiou, the continuum hypothesis is the most profound site of thought and being, the place where the two become most emphatically indistinguishable, and from which we can proceed with sound ontological investigation . . .

*Continued in Part II*