Which is the most powerful language, set theory or category theory? As far as I know, mathematics is written based on a language which can be for example set theory or category theory. My concern is about the power of these languages. How can we realize which language is more powerful and we can construct more structures with them? Maybe, there is an assumption which implies all languages are equal, and actually, there is no strength preference although there is maybe the simplicity preference.   If it is the case, could you please clarify why you think this assumption is good and obvious?
p.s. : I am not an expert, so if you think my question is absolutely wrong or does not make sense at all, please let me know.
 A: Set theory and category theory are both foundational theories of mathematics (they explain basics), but they attack different aspects of foundations. Set theory is largely concerned with "how do we build mathematical objects (or what could we build)" while category theory is largely concerned with "what structure to mathematical objects have (or could have)"?
Mathematicians work in informal set theory and informal category theory, which are immensly useful as lingua franca and as collections of universally useful concepts and techniques, but their formal versions are not actually needed by mathematicians for the most part. This is witnessed by the fact that the average mathematician is unable to list the axioms of Zermelo-Fraenkel set theory, and even of first-order logic. Yet, they are perfectly able to do complicated math.
The formal versions of set theory and category theory are of interest to people who study foundations of mathematics. These relationship between these two and computation has been known for a while. I highly recommend Bob Harper's blog post about the Holy Trinity for a quick read, and Steve Awodey's From Sets to Types to Categories to Sets if you would like know more about the connections and their significance.
The upshot is that we can mostly translate between set theory, type theory, and category theory, and that ordinary mathematicians could do their mathematics in either of the three systems, but the systems are not exclusive. In fact, a smart mathematician will be aware of their connections and will take advantage of them.
