In what sense is set theory the "meta theory" of analysis?

Question

Here, Terence Tao said:

I deliberately chose not to be excessively formal with regards to the set-theoretic foundations of mathematics here, which I am regarding as part of the metatheory of analysis, rather than part of the theory.

So basically he is saying that set theory is the metatheory of analysis. But in which sense is this correct? Checking Wikipedia: "A metatheory or meta-theory is a theory whose subject matter is some theory". In set theory, can one study the theory of analysis from a metamathematical point of view?

Answer 1

"Set theory" is a vague term, which may refer to "naive" set theory that involves basic properties of point sets, functions, relations, etc. These are at the same object level as analysis, although they are more generally applicable and are used also in geometry, arithmetic, algebra, etc. Tao specifically refers to "set-theoretic foundations" however. These are axiomatic theories that "interpret" analysis by reconstructing it from what is provided in the axioms themselves, and nothing else. The most popular of them, ZFC (for Zermelo-Fraenkel-Choice, the last one refers to the axiom of choice), formally builds everything from a single predicate (belongs to) and a few objects provided by its existence axioms (empty set, an infinite set, the powerset of a set, etc.). Some of the constructs are then identified with analytic objects.

Constructing a meta-theory usually means axiomatizing a field, and reasoning about what can and can not be derived in the resulting formal theory, e.g. whether it is consistent, complete, its axioms are independent of each other, etc. A standard way of resolving these issues is constructing various set-theoretic models, e.g. Gödel proved (relative) consistency of the axiom of choice with other axioms of ZFC by constructing ZF models where it holds, Hilbert similarly proved the indepenedence of the parallel postulate from other axioms of Euclidean geometry. There is a similar way of axiomatizing real analysis, and the study of such axiomatization(s) is accomplished by formally embedding it into axiomatic set theory, see Is the axiomatic approach to defining R rigorous? This is why Tao considers "set-theoretic foundations" to be "part of the metatheory of analysis".

Answer 2

The wikipedia article seems somewhat at odds with how I understand the term to be used — the metatheory provides the elements in terms of which we will discuss a theory.

I.e. set theory tells us about sets and functions and how to calculate and reason with them, then analysis uses the notions of set and function to develop calculus and such.

One could, I suppose, develop a field of "metaanalysis" which is the study of analysis itself, but that's not what's being implied by the term "metatheory" here.

Answer 3

Tao elucidates in a comment:

Strictly speaking, if one wants to discuss the theory of logical deduction properly, one should take care to distinguish between the “internal” formal theory under discussion (e.g. propositional logic, or first-order logic), and the more informal (and “external”) metatheory used to discuss that formal theory. With such a careful perspective, deductive rules such as “Given that “If X is true, then Y is true”, one can deduce “If Y is false, then X is false”” are part of the external metatheory, rather than the theory itself. Actually, strictly speaking, the use of phrases such as “is true” or “is false” are already part of the metatheory; if one were to adhere to the formal syntax of propositional logic completely, one should be instead saying things like “Given that “$X \implies Y$“, one can deduce “$\neg Y \implies \neg X$“.”

(I have attempted to use LaTeX, but I'm not sure if philosophy.se supports it -- hence the possible dollar signs in the quoted passage are mine.)

It seems that Tao uses "metatheory" as one would use "metalanguage" to discuss an "object language" (or, analogously, expound the "object theory"). Hence the term appears synonymous with "foundations of math".

Answer 4

Set theory is by no means the foundation of mathematics or analysis but in clear contradiction with it. Although set theorists choose to simply have a blind spot in this matter, it is clear to every objective observer. The simplest example even understandable to non-mathematicians is this:

When Scrooge McDuck daily receives 10 enumerated dollars and daily spends one dollar, namely that one with the lowest number, then he will get bankrupt in the set theoretical limit. In the mathematical limit he will become infinitely rich.

When McDuck spends always that dollar with the highest number in his possession, then he will become infinitely rich even in the set theoretical limit. This index-dependence shows that set theory cannot have any scientific application either.