I'm sorry but you misunderstand the situation. Set theory, type theory and category theory can be used for metamathematics [1] whereas real analysis and number theory cannot. They are foundational theories whereas real analysis and number theory are not. The reason for the difference has to do with the expressiveness of the formal languages and how they themselves can be expressed without a prior language other than formal logic itself.

[1]. https://en.wikipedia.org/wiki/Metamathematics

[2]. https://en.wikipedia.org/wiki/Foundations_of_mathematics

No more nonsense than studying English in English. It is true that type theory is mathematics, but the relevant fact is that it (unlike many fields of mathematics) is suitable for studying mathematics with.