Questions about the concept of Structure, Model and Formal Language

Question

When we start to define mathematical logic (specifically, propositional, first order, and second order logic) we start defining the concept of a language. At the begining this is done in a purely syntactical way. So far, so good.

Now, in this formalization we have been using what is called "metalogic" and "metamathematics" (we use sets, funcions, recursion, the process of reasoning, etc.). Then we want to attach the meaning of "truth".

(*)In the language of propositional logic we do this by means of the truth tables and the rules of deduction. In the case of first and second order logic we use the concept of "structure" (An actual set where the funcions and relations interact) and also the rules of deduction with quantifiers. Then it comes the concept of "model" and my problem.

As I understand, a model is a structure in wich a system of axioms, expressed in a given language, are satisfied.

Question: Am I right?

But in this case we have defined the term "structure" in terms of the metamathematical sense. So, when we say that we are making a model, we speak also in terms of metamathematics.

Now let's talk about the language of set theory. We built up this language to be sufficiently powerful to express all mathematics.

The axioms of $\mathsf{ZF}$ are then given. But then, to attach meaning to the expressions in the language, we actually need to suppose that there is a model of $\mathsf{ZF}$. It means we need to consider that there exist a structure that satisfies the axioms of $\mathsf{ZF}$ (of course in order to be consistent).

Assuming that the axioms of $\mathsf{ZF}$ are consistent, then from the language of set theory and the axioms of $\mathsf{ZF}$ we can build up all of the mathematics.

Question: given that practically everything can be constructed inside the Language of Set Theory (with $\mathsf{ZF}$) and we use it as such, why people say that it is a metatheory?.

Question: if everything can be expressed by means of the Language of Set Theory (with $\mathsf{ZF}$), Meaning most of mathematics we practically use, why to consider a different language? Like for example the language of group theory, the language of arithmetic, etc.

Also, when considering a new language, and a system of axioms, we need to talk about a model that satisfies the axioms(in order to be consistent). But such a model is a structure and people use the axioms of set theory ($\mathsf{ZF}$) to build it up. This makes me think of the next question.

Question: when we say that we are modeling, then are we working outside in the methateory or are we working inside the language of the set theory?

I really appreciate any comment about this topic and also any observation that helps me understand this concepts.

Answer 1

You're mistaken about second-order logic. It is just as a syntactical concept as first-order logic, but the logic itself is not as "nice" as first-order logic, and the idea of a second-order variable already refers, to some extend, to the idea of a set.

Question 3: You're half right. The rules of deduction, are in the language, those are syntactical rules because proofs are syntactical. But a model of a theory is an interpretation to the language, where the axioms are true.

Question 2: We refer to $\sf ZF$ as a metatheory because that is the framework in which we can develop logic (syntax and semantics together, even for "large" languages which include uncountably many symbols) and then we work on other theories inside the universe of set theory.

This is the theory of the universe, it's not the theory of calculus, of arithmetics, or groups. It's the underlying theory. And when we argue about groups, or about arithmetics, or about whatever, we argue in those relevant theories. And sometimes we make arguments about those theories, and an argument about a theory is an argument in the meta-theory (hence the prefix meta).

For example the statement: "Group theory does not prove the statement $\forall x\forall y(x*y=y*x)$" is a statement about the theory of groups, and therefore it is a statement in the meta-theory. Whether or not the meta-theory is $\sf ZF$, or something else, is irrelevant at the moment. But it's a statement about the theory of groups.

Question 3: Because the language of set theory is the meta-language. When we want to talk about the theory of fields we need two operations and two constant symbols. That's the language of fields. The language of set theory is the language of the underlying universe.

This question is the same as asking "If the CPU works with opcodes, why do we need C++, Java, Common Lisp, or Haskell?"

Yes, we can certainly express things with $\in$. But that would completely obscure any possible meaning of anything we would like to write, and you'd have to write incredibly long expressions for pretty much anything.

Question 4: If we agree that $\sf ZF$ is the meta-theory, then the arguments are essentially are in the language of set theory. Of course we don't write the in formal expressions using $\in$, but we can. It's just excruciatingly long, and if you're not using a proof assistant it's also useless.

This is why we have English.