0. Continuing the process of extracting and spelling out, in mini-steps, the theses operative in the thought of Alain Badiou: he tells us, still early on in his Being and Event (translated by Oliver Feltham (London and New York: Continuum International Publishing Group, 2005), p. 16), that:
…[Paul] Cohen’s concepts (genericity and forcing) constitute, in my opinion, an intellectual topos at least as fundamental as Godel’s famous theorems were in their time. They resonate well beyond their technical validity, which has confined them up till now to the academic arena of the high specialists of set theory. In fact, they resolve, within their own order, the old problem of the indiscernibles: they refute Leibniz and open thought to the subtractive seizure of truth and the subject.
It is with but a part of the last sentence of this passage that this post is concerned. I have included the first two sentences of the passage only to provide the referents of the “they” of the last one. I am, moreover, waiving for the time being any discussion of what he means by saying in the last sentence that “they [Cohen’s concepts of genericity and forcing] … open thought to the subtractive seizure of truth and the subject.”
My concern here and now is with Badiou’s understanding, if I read him aright, that Leibniz’s thesis of the identity of indiscernibles is refuted. (At least I assume that it is Leibniz’s thesis of the “Identity of Indiscernibles” which Badiou holds to be refuted and not Leibniz’s thesis of the “Indiscernibility of Identicals.” If my assumption turns out to be wrong, I will have to withdraw this post and write a very different one.) For if he holds Leibniz’s thesis of the identity of indiscernibles to be refuted, he should also hold it to be false.
1. The thesis, that is, of the “Indiscernibility of Identicals,” that:
For any existent x and any existent y, if x and y are identical, then any property P is a property of x if and only if it is a property of y.
is regarded by at least many, if not all, in philosophy as uncontroversial in at least many contexts, if not all. This is not the case, however, for the converse thesis, that of the “Identity of Indiscernibles,” that:
For any existent x and any existent y, if any property P is a property of x if and only if it is a property of y, then x and y are identical.
2. If, again, I read him aright, Badiou holds the thesis of the identity of indiscernibles to be false, then, by implication, he holds the contradictory thesis to be true, that:
It is not the case that, for any existent x and any existent y, if any property P is a property of x if and only if it is a property of y, then x and y are identical.
Now this in turn is logically equivalent to:
For some existent x and some existent y, it is the case both that any property P is a property of x if and only if it is a property of y and that x and y are not identical.
In other words, Badiou is committed to the certainly controversial ontological thesis that there are at least two existents x and y which, though similar in all respects and thus not at all different, are yet distinct and non-identical; they are precisely indiscernible.
3. He seems thereby to be equally committed to other, less controversial theses. For one thing, given that Badiou holds that there are at least two existents x and y which, though similar in all respects and thus not at all different, are yet distinct and non-identical, he should logically also hold, against ontological monism and with ontological pluralism, that:
There are at least two existents.
Again, he should logically also hold, against ontological nihilism and with ontological realism, that there is at least one existent, that there is at least something, or that.
At least something exists.
4. I will finish by taking two mini-steps “upwards,” one into Badiou’s theory of truth and the other into his theory of knowledge. First, then, given that he holds that there exists at least one existent, he should also hold that:
The proposition, that at least something exists, is true.
He should, moreover, also hold that, I will put it:
The reality (or fact), that at least something exists, is known.
What the import of these two theses might have will have to await an understanding of Badiou’s theories of truth and of knowledge beyond that which I have at present.
After Aristotle 
I think you may be mistaken about the Id. of Ind. principle in Leibniz. He would not want this to imply the existence of any objects. Rather, if there exists an x and a y, if x and y share the same properties and nothing besides, then x and y denote the same individual.
Thus, by rejecting it, one would be merely accepting the possibility that x and y share the same properties and none besides, and yet denoting two distinct individuals.
Leibniz’ challenging question to this video would be: how can you distinguish two sets of properties, without grounding it in some difference in the properties?
I think Id. of Ind. stands or falls with one’s concept of individuality: What makes a [singular] individual? The conjunction of properties? Then the principle holds. Haecceity? Then it holds not. Some other principle of individuation?
I’m not sure whether or where we disagree. My concern is not really with Leibniz, but with the principle of the “Identity of Indiscernibles” and Badiou’s thought that it had been refuted. My first question to you, then, is that of whether or not you find acceptable the formulation of the principle I had in mind, that:
For any existent x and any existent y, if any property P is a property of x if and only if it is a property of y, then x and y are identical.
If you find it acceptable, then let’s turn to Badiou. If I am right that he holds that it is this principle which has been refuted and that that entails that he holds or should for consistency’s sake hold that it is, as refuted, false, then it seems to me that he is committed to the thesis contradictory to that of the “Identity of Indiscernibles,” that:
It is not the case that, for any existent x and any existent y, if any property P is a property of x if and only if it is a property of y, then x and y are identical.
Do you find this set of steps acceptable? If so, then, am I right in seeing the thesis just given as logically equivalent to the following?
For some existent x and some existent y, it is the case both that any property P is a property of x if and only if it is a property of y and that x and y are not identical.