From the perspective of these two philosophers there's not a debate about whether set theory or differential calculus are better, more important, more appropriate or anything in general terms. The discussion framing the book mentioned by DeLanda revolves around metaphysics and the specific claim made by Badiou that Set Theory was sufficient to found what Deleuze would call "any multiplicity whatever" covering all classes of multiplicity. If the two mathematical references are posed in any sense against each other, it is due primarily to the reception of a particular framing set up by Badiou of his opposition to Deleuze in which Deleuze is posed by Badiou simultaneously as a philosopher of "the One" that only appears to promote a philosophy of difference and also as his biggest enemy because despite being a philosopher of "the One" (his term), he fails to recognize Set Theory as the basis for philosophical ontology, (which ultimately shapes the differences in their conceptions of what philosophy is/does). In the framing of this opposition by Badiou the universality of the Set is perhaps framed against the crowned anarchy of difference for which differential calculus played a role in the enunciation of the case presented in Difference & Repetition, but this perception of some opposition is the result of the dust blown up by the polemic, or from the brief mention in Clamor of Being where Badiou states that upon their meeting Deleuze was occupied with this set of mathematical problems and he with that. But there is no actual opposition of these mathetical references as suggested by DeLanda'a quote. It is key to note that both shared similar views on the role of mathematics in the formalization of problematics independently of the other, and perhaps that is what initially attracted Badiou to Deleuze's work despite not being of his camp. The differences, which seem larger than they really are (such as if you were to compare either to two other names in philosophical history) are primarily political --this having a number of consequences which people have to understand in order to properly contextualize the absolutely crazy history of Badiou's historical antagonism of Deleuze; and secondarily their different approaches to and perspectives on ontology.
And before moving on, I would say that DeLanda's characterization of the book as "incompetent" misrepresents what Badiou is doing. Anyone who reads Badiou can see an extremely sharp mind at work, it's not an issue of competence (and his book on Wittgenstein can be read for comparative purposes on this point). His book contains eggregious falsifications and misrepresentations (from the non-Badiouan perspective), not because of incompetence but because his politics demands it. The product of his work remains incomprehensible until you understand his politics IMO. Please see Badiou's "Politics as Truth Procedure". (I'd be happy to expound on this and it's relevance to the question, but I didn't want to go into it here for fear this would be extremely long)
Direct answers to your questions:
Is the differential calculus reducible to set theory? And how? And who
came up with the needed theory?
In just about any discussion of these two the answer is clearly yes. But one should be clear about what the question behind your question is. To philosophy the question seems to be more about the specificity of the tools appropriate for a particular Idea or problem (allow me to make clear that the 'or' here isn't synonymizing). If Deleuze in his hat of historian of philosophy is retracing the steps of Leibniz it would make no sense to discuss set theory in lieu of calculus. He would simply say that both play different roles in the formulating of "problems" for philosophy. In Badiou, where set theory plays the central role in formal thinking of entities, the question takes on importance and the question of reducibility is key, but in his framework of thinking alongside Deleuze this is not ever an issue of contention, and that's because the discussions (around the particular problems being confronted philosophically) are different. He reads Deleuze's univocity thesis in the same terms as he develops his own Platonic theory of multiplicity. A theory of mulitplicities and one of differentials are directed towards two different planes/problems.
Is set theory more related to limits while the differential calculus
is more related to infinitesimals?
With regards to set theory it's not simply a concern about axioms and limits but of the meaning of "being" in the most basic terms or of the most general and abstract formal and structural conditions for anything to exist at all. For Badiou, his development of a set-theoretical ontology was to establish general formal conditions for the consistent presentation of any existing thing, "counted-as-one" and as a coherent unit. Whereas being in itself is simply "pure inconsistent multiplicity" or multiple-being without any organizing structure.
So there's a number of things he attemped to derive from this framework, but the two most central aspects are:
1. a priori conditions of any possible ontology
2. a theory of pure multiples (including a typology of multiples, and implications for what this means for truth, Events and singularity)
Whether he failed or succeeded or not in Being & Event, a proper way to assess the issue of his formalization was as a measure to rigorously define a space of action and creativity as opposed to simply be controling/limiting operations.
With respect to differentials, they can't be reduced to infinitesimals. Insofar as infinitesimals were invoked in Deleuze it was to demonstrate how Carnot and Leibniz demonstrated how problems resist being absorbed by solutions.
If Badiou is more into set theory, and probably monads, is this closer
to limits? It seems for me that there are big implications in seeing
the world as objects and not as differences or intensities.
With respect to the differences between objects and intensities, there were major limitations to the framework Badiou developed in Being & Event, but by Logiques des Mondes Badiou adopts Deleuze's understanding of intensive difference pretty much to the letter. Also Badiou's work expands a lot beyond his philosophical formalization of set theory, so if we were to do him justice here we would also have to include all of the developments he made around category theory.
How are set theory and the differential calculus related? Did Deleuze
try to revive an alternative interpretation to make use of another
branch of the scientific tree to open up room for different
epistemological interpretations (a world of intensities in becoming)?
Are these maybe better discussed in the form of rates of change
(differential calculus) rather than sets (definition of borders and
separation of objects).
"What matters to us is less the determination of this or that break [coupure] in the history of mathematics (analytic geometry, differential calculus, group theory ... ) than the manner in which, at each moment of that history, dialectical problems, their mathematical expression and the simultaneous origin of their fields of solvability are interrelated." - D in Difference & Repetition p. 180
Others can explain the relations between set theory & differential calculus in mathematical terms more concisely than I, so let me focus on your second & third questions. If we approach these passages as attempt to found alternative epistemological interpretations it obfuscates the DIRECT objects of these works. Deleuze had a great interest in mathematics and "hard" sciences, but equally in arts, music, linguistics, anthropology etc. Generally if something specific in a field is being addressed it's at the service of some larger target (e.g. making clear semiotic operations of deterritorialization from reterritorialization, to show what role axiomatization plays vis a vis a-signifiying ruptures etc, which as semiotic concerns, are not restricted to any particular field or discipline, thus you'll see an example as related to science here, another related to music there, another related to economics over there). When Deleuze revisits the issue of the development of the thinking behind calculus in Leibniz (both in Difference & Repetition and in his stand alone book The Fold: Leibniz & The Baroque) it's first to elucidate the complex web of concepts surrounding Representation, Faculties, Ideas, Problems, Solutions, what it means to think, and the nature of differentials as unsubsumable to solutions; the second use is to trace the development of an Idea (specifically differential relations) via the problematizations of thought actually ran through cases of philosophical preocupations (regarding perception, extrinsic physical causality, affective qualities) to the "convenient or well-founded fictions" (as Leibniz described his calculus to Varignon) and extensive reality. In both cases, while Badiou is correct in saying that Deleuze was extremely interested in mathematics, it's not because he is concerned with what mathematics does with these problem, the development he makes of them is within philosophy. Alternative examples can be seen in something Badiou would equally criticize him for, writing books invoking hundreds of cinematic cases that are of no use to cinematographers or film theorists, but rather to make statements about movement, sensory-motor schemas and time, a book about a painter that is really about the relationship between thought and sensation. Badiou has a different agenda that of course priviledges the mathematical that really is of no interest to Delezue. But the point here is that there are no epistemological direct motives here, there may be epistemological consequences, but they would be secondary to the things concepts and worlds that Deleuze was actually focused on creating.
