Set-theoretic notion of differentiation and integration? Since set-theory is said to be one of the foundations of mathematics, I would think that every mathematical concept is definable in set-theoretic terms, right? 
How would you define differentiation and integration? How would you define the limit concept? 
Thanks.
 A: Once you have the structure of the real numbers defined, the definition of integration and differentiation is the same as you would expect it. We have a language in which we define integration, we just have to interpret it into set theory.
The real numbers can be constructed in several ways in set theory (the important part is that we can prove that all the constructions are isomorphic to one another).
Say we take the Dedekind cuts definition. Then from the rational numbers we can construct the real numbers, and we can extend the operations of the rational numbers to those of the real numbers. So the question is how do we define the rational numbers?
Well, we can define the rational numbers as the field of fractions of the integers, and the operations of the integers extend naturally to those of the rational numbers. So we really just need to know how to define the integers and their operations.
But the integers are really just the completion of the natural numbers to allow additive inverses. So if we know how to construct the natural numbers in set theory, then we can certainly construct the integers, and the operations will extend.
And indeed the natural numbers have a relatively simple construction (there are several construction, this is just one). We define $0=\varnothing$ and $n+1=n\cup\{n\}$ (so we have that $n=\{0,\ldots,n-1\}$. Then we define addition and multiplication as cardinal arithmetic, $m+n=k$ if and only if $k$ have the same cardinality as a union of disjoint copies of $m$ and $n$; and similarly $m\cdot n=k$ if and only if $k$ has the same cardinality as $m\times n$ (the Cartesian product).
Now we have the natural numbers, their operations. We can start constructing sets. We construct the integers, then the rational numbers, then the real numbers. Now we have sets $\Bbb R$ and $+^\Bbb R$ and $\cdot^\Bbb R$, and we can define the rest of analysis from these sets, as if they were the "true" operations of $\Bbb R$.
And eventually, we have a definition for convergence, then for continuity, differentiability and differentiation, and integration. Those definitions are long, complicated, and uninteresting, and if you wish to "unwind them" then they become very dependent on the sets $\Bbb R$, $+^\Bbb R$ and so on.
But those definitions exist. Just very very very very long, and very very very uninteresting. The interesting part is that we can how they exist. And so we have. By showing that we can just recreate the basics, and that all the definitions follow.
