Domains in First-order logic

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?

1. 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!

• The domain of a theory is a set in whichever metatheory of sets we use. As in all fields of mathematics, in logic we can be a bit ambiguous when it comes to our ambient set theory. So when we say "Let $G$ be a finite group" or "Let $n$ be a natural number", we are a bit ambiguous as to what these mean. I would say that something similar happens in logic: We say "Let $A$ be a set". You can think that this set is a particular object of the universe of set theory that we "live in" when doing mathematics, or you can think of this as something that we proved that exists in our set metatheory. – Apostolos May 24 '17 at 0:00
• You wrote "the universe of set theory that we 'live in' when doing mathematics". I view proving theorems in mathematics (eg doing analysis) as reasoning which could be formalised in first-order logic (let's forget about other logics for now) where our theorems then could be derived in natural deduction (henceforth, nd) from the axioms of ZFC. In some sense, I view mathematics as "reasoning in an arbitrary model of ZFC" (so completeness guarantees existence of derivation in nd), is this a reasonable view? – SimonSimon May 24 '17 at 0:40
• In your metatheory ZFC-based you want to define a domain $A$ for e.g. your f-o theory of arithmetic. Then you pick up the structure $\mathbb N$ that we have proved to exists by ZFC. What is $\mathbb N$ ? Is a set: the set that ZFC has proved to exists and that satisfy certain axioms and theorems of ZFC. Then we assume it as domain of interpretation of the theory and its elements we call them: numbers. – Mauro ALLEGRANZA May 24 '17 at 6:13
• Of course, we can consider domain also in a more general sense, according to the fact that we can "use" formal logic also in non math contexts. Consider a trivial example based on Aristotelian syllogistics: "All men are mortal." etc. In this context, a suitable domain is the "collection" of all living creatures. – Mauro ALLEGRANZA May 24 '17 at 6:42
• NO; $\mathbb N$ is the "formal" counterpart (in the meta-theory ZFC-based) of the "usual" natural numbers. – Mauro ALLEGRANZA May 24 '17 at 11:21

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.

• I think I see your point, but both $\mathbb{R}$ and $\mathbb{N}$ can be constructed as sets in ZFC: When doing ordinary mathematics, I would definitely write the example as you wrote it. However, given that my "background theory" is ZFC, isn't it implicit that every such example could be rewritten in a way such that the quantifiers range over all sets (in my domain of discourse)? In your example, I could write $\forall x$ and $\forall n$ (quantifying over all allowed sets) and then move $x \in \mathbb{R}$ and $n \in \mathbb{N}$ to the antecedent of the conditional? – SimonSimon May 24 '17 at 10:38
• @Mauianon If you are interested in foundational issues, these distinctions may be important (can't comment). In ordinary mathematics, however, they may be an unnecessary distraction. I don't think you will ever see mention of a domain of discourse in texts on number theory, analysis or algebra. Also, I see $\forall x\in R:\cdots$ as simply a shorthand for $\forall x: [x\in R\implies\cdots$. I have found that some manipulations are easier to understand using the latter form. – Dan Christensen May 24 '17 at 19:35
• I am interested in foundational issues, specifically how ordinary mathematics relates to set theory and philosophy. Though to obtain some understanding, I try to start with the more technical questions before I get lost in the philosophical ones :) – SimonSimon May 24 '17 at 19:53
• @Mauianon Don't know if this is relevant, but in my day, you could get an undergrad degree in pure math (I didn't) without ever formally taking a course in set theory, logic or philosophy. – Dan Christensen May 24 '17 at 20:22
• Yes, that is more or less my situation: BSc and future MSc in pure math with minimal exposure to logic etc. I guess my lack of exposure to these subjects is what makes them difficult for me to understand. My logical / philosophical concerns are purely out of interest, they do not really matter in practice for my actual math courses just as you've pointed out. – SimonSimon May 24 '17 at 20:31

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).