Badiou Explained in 1000 Words

Badiou’s philosophy rescues ontology form being either a pseudo-problem (Wittgenstein) or a tragic story of finality and withdrawal (Heidegger).  To do this Badiou solves the basic problem of the question what makes a thing the thing it is, which has been handed down to us from the Greeks.  The totality of ontology is basically contained in Aristotle’s consideration of categories through the idea of divisibility and commonality or species and genus.  To work out what a thing is you need to name its ontological specificity, this is a closed off thing, and its ontological generality, this is a single example of all such things.  Thus the specificity of a thing is its species, and its generality is the genus.  This should allow us to differentiate one thing from another in terms of what is unique to the thing (species) and what it shares in common with others (genus).
Problems always arise at the upper and lower limits of course.  What, Aristotle asks, is the genus of the genus or the maximal upper limit?  Traditionally this is handed over to an inconceivable and thus humanly inaccessible infinity that is essentially the province of God.  At the same time when do you stop in terms of specificity or what is the absolute lower limit of the species that cannot be the genus of another species?  Traditionally this is handed over to quasi-natural concepts of vitality, substance or atomism.  The history of Western thought has been dominated by three insoluble paradoxes: the genus of the genus, the species of the species and their inter-relation.  Most systems have solved these by installing God at the top, nature or life at the bottom, and a fudged compromise in the middle. Kant is the best example of this.
Badiou’s intervention is amazingly simple.  He discovers that a key branch of modern mathematics, set theory, especially that of Cantor, solved the problems of inaccessible infinity and infinite regress through two axioms.  Badiou defines the greatest idea of our age as the laicisation of the infinite through Cantor’s discovery that while we cannot count until infinity, infinite numbers are numbers and you can calculate with them.  Thus there is an upper limit or a genus which proceeds from no species. 
At the other end of the spectrum Cantor also found a halting point for every species which is the void set.  This is a set to which nothing belongs.  That said he was also able to prove that the void set belongs to every set, which means it also belongs to itself.  Thus zero or the void is a calculable halting point at the lower level beneath which are simply more voids.  However many voids are in the void set, as they are all equal to each other, the void set cannot be made any smaller.  Badiou summarises this in his simple but brilliant ontological axiom: being is not.
If we summarise, Badiou is able to impose an infinite that is actual, a void which is present as absent, and thus a means of counting something as one without making it the One, or being forced to further divide it into two smaller ones.  Solving the problem of infinity and regress, Badiou revitalises ontology and the metaphysical project.
What interests Badiou however is not the ontological consistency that this establishes, he takes this as read in an axiomatic and intuitive fashion.  Rather he sees that the consistency of states is always based on a radical inconsistency at its upper and lower levels.  When a state is established then ones are counted twice.  First, at the ontological level a set counts as one in terms of the elements which belong to it.  Second, at the level of our world the state counts the count and defines a stability in the world by imposing a limit to the count.  The first count defines what belongs to a state, the second differentiates what belongs from what is included or sets and their subsets. 
What Badiou realises is that the first count must include at least one element that it cannot count, the void set.  This void is real because without it the set cannot be counted without the problem of infinite regress, thus it is inexistent or exists as something which cannot be counted.  This means every count is founded on a dependence on a void element and this void element is real so that in certain circumstances it can present itself.  In terms of the second count Badiou notes the axiom of the power set which says that the set of the total amount of subsets in a set is always larger than the set itself and operates in effect as multiple of multiples of an actual infinity.  The state, aware of this problem, imposes a second count and determines that only certain elements will be counted, while the rest will be included in the set as subsets that do not belong or are not counted.  Again Badiou notes that the state is also founded on the inclusion of a destabilising element, in this case the potential infinity of elements or subsets that could be counted but are not.
Badiou’s task is simply to find a means by which the void can be made to link up to the uncounted subsets and his theory of the event is precisely that.  The event is a void that presents itself to a situation as present but not included whose impact is assured by a fraternity of militants in any situation who feel they are included but do not belong.  They become champions of the event and in such a way their actions become operative truths in a new situation.  This is the totality of Badiou’s system, how to stabilise the system using the axioms of the void set and the power set, and then ironically how to destabilise the system, precisely because of the same two axioms. 
Its elegance is truly a thing of power and beauty.