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:
- 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?
- 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!