Domains in First-order logic

Question

I’m a master’s student in mathematics with two questions about logic. I have taken a course in first-order logic (basically covering soundness and completeness) a few years ago, but I have to admit that my understanding of logic is weak.

These are my questions:

1. In first-order logic, an interpretation $\mathcal{A}$ consist of a ”non-empty universe of discourse” $A$, together with some functions and relations on $A$. Formally, how do we know which objects qualify as ”non-empty universes of discourse”?

I will call $A$ the domain of $\mathcal{A}$. To answer my own question 1, I guess which objects qualify to be used as domains is decided by what metatheory we are using: For example if we work with models of ZFC, we typically want our domain to be something like the von Neumann universe $V$. To be allowed to use $V$ as the domain of our model, we use ZFC together with some additional axioms as our metatheory, probably to guarantee to "existence" of $V$. Is this answer to question 1 correct?

2. If the domain $A$ of my interpretation $\mathcal{A}$ happens to be a set of ZFC (say $A = \mathbb{N}$, the natural numbers as constructed in ZFC), then exactly what object does $A$ refer to? In lack of better words, does it refer to the ”syntactic” set $A$ which I can prove the existence of in natural deduction, or does it refer to some ”semantic” set $A$ which comes from some model of ZFC? Does this question even make sense, or does it reveal some misconception I may have?

Thanks in advance for all help!

Answer 1

In mathematics, we don't usually have "domains of discourse" since we may have multiple [edit: possibly empty] domains even without the same statement. As such, they are usually made explicit for each quantifier.

Example: $\forall x \in R: \forall n \in N: [\neg [x=0 \land n=0] \implies x^n \in R]$

In this example, we have two domains: the set of real numbers $R$ and the set of natural numbers $N$. In this case, it makes no sense of talk of a single domain of discourse.

Answer 2

As to your first question the situation seems quite clear in the context of ZF. According to the axiom of infinity there's at least one (inductive) set. So, by separation and extensionality there's exactly one empty set, $\emptyset$. The condition $x\not = \emptyset$ is certainly definite.In this way we obtain the proper class of non-empty sets. So, empty sets and thus possible universes of discourse are exactly the members of that class.

Regarding your second question the distinction you're making between syntactic and semantic sets is unusual from a set-theoretic perspective. After all, in ZF $\mathbb N$ is introduced as a name of the least inductive SET and thus denotes, well, a set. The syntactic view seems to be grounded in the belief that we have a complete proof theory for ZF. No we don't, due to Gödel's first incompleteness theorem (at least if ZF is consistent).