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

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.


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.

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.

Tags:

Logic

Set Theory

Soft Question

Foundations

Meta Math

Related

Why does Friedberg say that the role of the determinant is less central than in former times? Prove that $G$ is an abelian group if $\{(g, g):g\in G\}$ is a normal subgroup. There must exist a random variable with certain given law? If $p\geq 5$ is a prime number, show that $p^2+2$ is composite. Prove that for any set of 5 integers, there is at least one subset of 3 integers whose sum is divisible by 3 Any way to solve this right angle triangle problem without trig? Determinant of the derivative of a map between manifolds Gradient of Sobolev function on level sets Why does a unitary in the Calkin algebra always lift to an (co-)isometry? If $f$ is differentiable on $[1,2]$, then $\exists \alpha\in(1,2): f(2)-f(1) = \frac{\alpha^2}{2}f'(\alpha)$ Show that $f(x)=\pi-2\arctan(\sqrt{x-1})$ Any more cyclic quintics?

Recent Posts