So you don't wind up saying something like this:
Broken into 37 meditations, Being and Event is centrally an intervention in what Badiou calls the “Cantor-event.” It goes something like this: Georg Cantor’s work in set theory circa 1874 shatters the distinction between the finite and the infinite by proposing that in any given set of numbers, say [a,b,c], the one, a, is merely a count and not oneness in and of itself. Rather, it is an effect of the presentation of the multiple, a, b, and c; all three take place in the particular situation of the set. Such a presentation allows for the members of the set, and not vice versa. Badiou uses set theory to revise the Heideggerean being-as-one: “Ontology, if it exists,” he says, “is a situation,” that is, one in which beings-as-multiples are presented. It is this structure of which a representation of oneness is an effect.
And his startling proposition: ontology, if it exists, is mathematics.
There's definitely something startling here, but it's not that proposition. Rather, it's the fact that Pythagoreanism is being cited with tacit approval. But the point of this post is not to dismiss the notion that the only fundamental kinds of things which exist are mathematical entities. Oft times those of pseudo-intellectual persuasion invoke scientific principles or mathematical theorems with the mistaken belief that these principles provide them with novel arguments for some thesis. This is almost always not the case. The Heisenberg Principle is likely the most abused scientific proposition, but here set theory is the whipping boy.
Cantor did not shatter the distinction "between the finite and the infinite". In fact, under any natural understanding of what it might mean to "shatter" such a distinction, there is no longer any distinction between the finite and the infinite. That, of course, is false. This post contains a finite number of letters.
What Cantor did was to introduce the notion of equicardinality between sets, such that set A and set B have the same cardinality if and only if there is a 1-1 function from A onto B. A function is a set of ordered pairs R such that, for all x, y and z, if <x,y> and <x,z> are members of R, then y = z. (Think of a set of points. The requirement for a set of points constituting a function is that for any x value of the function, there is only one y value. For the intuitive graphical examples, note that a vertical line is not a function, a horizontal line is a function, and a diagonal line is a one-one function.) Finally, and to fully define all the terms used in the definition of "equicardinality", a function with domain A is "onto B" just in case the domain of the function is identical with B.
Here's a nifty way of thinking of equicardinality that was recently suggested to me. Imagine a table at your favorite fine restaurant with dinner service set for eight. Suppose you wonder whether the table setting is complete, or perhaps, whether everybody had a wine glass. There are two obvious ways to figure out whether or not everybody has a wine glass. The slowest way would be to count all the chairs first, count all the wine glasses second, and then see if your numbers match. Most of us, however, would probably instead check to see if every chair is paired with a single unique wine glass. If you can match every chair to a single unique wine glass, then there is a one-one function from the set of chairs onto the set of wine glasses. That is, the sets have the same cardinality. For any finite number of chairs and wine glasses, these two methods of counting will return the same answer, but matters become tricky if we move to infinitely long tables.
There was a distinction between the finite and the infinite both before and after Cantor. But after Cantor, we could draw further distinctions between infinite sets on the basis of their cardinality. (It should be remembered that cardinality is a semi-technical notion, stipulatively defined.) Consider two infinite sets: the set of natural numbers {0, 1, 2, 3, 4...} and the set of even numbers {0, 2, 4, 6, 8...}. Intuitively, the set of even numbers contains exactly half plus one as many members as the set of natural numbers. But the two sets have the same cardinality. Each member of the evens can be paired with exactly one unique member of the naturals. That is, there is a one-one function from the evens onto the naturals. To see this, pair zero with zero, one with two, two with four, three with six, etc...
The notion of cardinality is, pretty obviously, not the same as the commonsense notion of size. Thus, the fact that two infinite sets have the same cardinality does not entail that they have the same number of members, though the fact that two finite sets have the same cardinality does entail this. (The former claim is mildly controversial.) But in any case, cardinality is a technical notion, and while applying it to infinite sets yields amazing mathematics, the results are not really surprising since "cardinality" is a notion introduced by stipulative definition. You shouldn't be shocked to hear that the set of the evens has the same cardinality as the set of the naturals, though you might well be shocked to hear that these sets are the same size, or have the same number of members, etc... With the notion of cardinality defined above, it can be proven that some infinite sets have greater cardinality than others. So while Cantor opened up the field of transfinite mathematics, but, not to belabor the point, shattered no distinction between the finite and the infinite.
I have no idea whether Badiou makes such a ridiculous claim or whether Alexandra Heifetz invented it, but either way, some training in analytic philosophy will prevent one from making such bizarre assertions. I leave to my readers the task of attempting to come to grips with the rest of that strange business involving sets not being "oneness in and of themselves".
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