Why does Tao use the word *metatheory* in this context?

Question

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?

Answer 1

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.

Answer 2

Wikipedia's page on "Meta" gives a better description:

Any subject can be said to have a meta-theory, a theoretical consideration of its properties, such as its foundations, methods, form and utility, on a higher level of abstraction.

An example of "foundations, methods, and form" would be sets and how to manipulate them and reason with them.

Answer 3

I think that you can compare with : Terence Tao, Analysis I (3rd ed, 2016):

• Ch.2 [page 15] : the natural numbers, defined in terms of Peano axioms

• Ch.3 [page 33] : set theory : "almost every other branch of mathematics relies on set theory as part of its foundation"

• Ch.4 [page 74] : the construction, using set theory, of other number systems : integers and rationals

• Ch.5 [page 94] : the construction of the real numbers.

Finally :

• Appendix A : Mathematical Logic : "which is the language one uses to conduct rigorous mathematical proofs."

Now, you can read them in reverse order : within math log we define language (and the tools) of first-order language with equality.

This is used to build-up (first-order) set theory.

With the concepts (and axioms) of set theory we may develop the number systems, up to analysis.

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The word "metatheory" is not uswed in the book; thus, I think that in the statement you are quoting, Tao means the "foundational framework" of real analysis : set theory formalized with first-order language with equality.