Why does Tao use the word *metatheory* in this context? In a comment under his page about the book Analysis 1, Terence Tao writes:

If one were to formalise the metatheory implicitly used [in the text Analysis 1], though, it would be a set theory as a language with equality

I am confused about his usage of the word metatheory here. Checking wikipedia, "a metatheory or meta-theory is a theory whose subject matter is some theory." But this seems to me to not be meant by Terence Tao, since the set theory described by Terence Tao in his book is not intended to be a metatheory of some other theory, it is just used as a basis from which one can rigorously define the objects one needs in analysis.
Do you know why Tao used the word metatheory then? Does this word maybe have another meaning than that described by the wikipedia article https://en.wikipedia.org/wiki/Metatheory?

The theory of the real numbers can be described on its own terms without being a part of set theory. Then you can use set theory as a meta-theory without altering the real numbers. For instance you probably didn't generate your real numbers up from the null set, you just started with some axioms about equations. Note that many times in courses there are "illegal" moves when you go outside the theory you are working on and use set theory.
