Why can't you formulate Voevodsky's 2-theories in set theory if its the foundation of maths? I was reading up on the motivations of Voevodsky's homotopy type theory and was puzzled when i read that his main motivation was for his work in infinity groupoids. 
He was working on what he calls "2-theories" and realized that he couldn't create them or even have derived them had he worked with the accepted foundations of math with set theory. He actually mentions lower dimensional categories as a foundation as well and how to a lesser extent they were also inadequate but not completely unable to derive the results. (Higher category theory works just fine however and indeed is how he derived these incredibly abstract ideas.)
But, how can a foundation of math not be able to derive something... in math? It wouldn't be a foundation at all then, wouldn't it? So far in my studies, i can confirm all of undergraduate mathematics can be said in set theory and category theory. And it seems the next step for even more complicated and especially abstract maths is higher category theory. Is it not time to evolve? 
*Voevodsky mentions that getting over (not higher) category theory as a foundation for him was the most challenging.
 A: I found an article that sounds like what you may have been reading. So this answer is based on that.
In Voevodsky's case, you are asking the wrong question. If you ask, "could Voevodsky's stuff in principle be built in set theory," the answer is probably, "yes."
However, Voevodsky wanted to do computer verification of his ideas, because they are such complicated constructions that he wanted assistance in not forgetting about the details. So, at that point, you must ask whether mechanized set theory is an adequate substrate on which to practically build a system for talking about these ideas. Are you actually willing to write out the ZFC proofs in enough detail, or do the tools allow sufficient abstraction that the job is feasible?
Voevodsky's answer was that set theory is not adequate in this sense. Instead, he worked on a foundational system that could more directly encode the "higher dimensional" things he wanted to talk about. Then, if this system is implemented in a computer, it is a better basis for the reasoning he actually wants to do. But, the world of set theory can be thought of as sitting inside it as the '0 dimensional' things, so it's conceivable that it could serve the purpose of existing set theories as well.
I would suggest that this is the real purpose of 'foundations' to begin with. There is nothing more justified about them, and other mathematical domains don't need to be justified by reduction to some foundations. The purpose of a foundation is to serve as a ready-made environment for building and exchanging ideas from various particular domains of mathematics. But, to serve this purpose, it must be convenient to reconstruct these other domains. So, it's not that set theory 'does not contain' the structure Voevodsky was interested in. The problem is that it is too inconvenient to construct it (in sufficient detail) in set theory.
