I know that there are attempts to do category theory in the framework of set theory, but I'm not asking that. I'm asking about the converse.
I will explain my thought below. I think my thought makes sense, but it's probably because I don't know serious logic. I want to know where I have false thought.
In set theory, we consider elements of a collection and every set theory textbook explains that set is a collection and it is a undefined term.
In set theory there are two undefined terms, namely "set" and "$\in$". We use these two terms to make an analysis on elements of a colletion named set and that is indeed our goal in set theory. We can think this as: "There are one undefined subject(set) and one undefined verb($\in$) in set theory".
We can restate "choosing axioms for set theory" as "choosing subject nouns in a language".
Hence, ZFC can be viewed as "a language consists of subjects(chosen by axioms) and verbs(first-order logic and $\in$).
Now, let's consider a situation that a foreigner named Category theory analyzing this language who knows only verbs of this language. (Namely, first-order logic and $\in$) So that this guy is not creating subjects of this language.
Say, the subjects he can only use are object,arrow etc. These words are completely different from those in ZFC. Hence, there is no worry about restricting words of this guy to avoid paradoxes like Russel paradox since, to him, subjects like $\{x:x\notin x\}$ are not given. Moreover, since category theory is not creating bigger objects from given objects, there is no size issue.
To sum up, ZFC can be viewed as a factory producing "subjects" and Category theory can be viewed as a councilor analyzing those "subjects". Hence, ZFC set theory can be embbed into Category theory without any contradictory concept.
Where am I thinking wrong?
