How are the axioms defined in the fundamentals of set theory

Question

I am reading Enderton Elements of set theory and something keeps bothering me. At the start he defines the axioms in terms of the symbols $$\forall, \exists, \in, \neg, \&, \text{ or }, \iff.$$ Then he goes on to define functions as specific types of set. However, it feels to me that the axioms and the theorems are sentences in some kind of formal language. And to evaluate the truth of a statement, means to define an interpretation of the statement in the 2 element set $$S:= \{\text{true, false}\}.$$ For instance, $$x\iff y$$ derives a function on the set $S$ such that $$\left((x\iff y)_S = \text{ false}\right)\text{ iff } \left(x_S = \text{ true and } y_S = \text{ false}\right).$$ My issue is how can this be if you need the axioms to define functions in the first place. It feels like the foundations of mathematics are self referential?

I appreciate my question is perhaps unclear, but I feel my understanding is too limited to ask a more precise question at this stage.

Yes, axioms of e.g Zermelo-Fraenkel Set Theory are formulas of predicate logic. See e.g. the Null Set Axiom: $∃x \forall y ¬ (y \in x)$. — Mauro ALLEGRANZA, Dec 21 '22 at 12:01
Thus, theorems of set theory are formulas expressed in the language of the theory that are provable from the axioms of the theory using the rules of predicate logic. — Mauro ALLEGRANZA, Dec 21 '22 at 12:03
You examples are not exactly formulas of predicate logic: and example will be the following definition: $\forall x [x \in A \cup B \leftrightarrow (x \in A \lor x \in B)]$. — Mauro ALLEGRANZA, Dec 21 '22 at 12:04
See also the post Axioms of set theory and logic as well as Axioms of First Order Logic and ZFC Axioms and Are "formulas" in Axioms of ZFC indefinite? — Mauro ALLEGRANZA, Dec 21 '22 at 12:07
functions in set theory are sets, i.e. sets with a specific property. See the post Exact definition of a function. — Mauro ALLEGRANZA, Dec 21 '22 at 12:12
It sounds like you're charging the axiomatization of set theory as circular: the formal axiomatization of sets (or at least its semantics) requires the very notion of a set. This is a common sentiment/worry, and yes, there is a notion of set that we already have in mind and kind of have to accept as a given when doing this kind of axiomatization. But note: of course we have to start with something as a basis: we can't be asking "But what's the justification/definition for that?!" forever. — Bram28, Dec 21 '22 at 14:46

Karl · Accepted Answer · 2022-12-21T15:37:36.800

You can formally describe the language and rules of predicate logic using set theory, but that formality isn't necessary in order to use predicate logic, as long as we have a shared understanding of the rules for forming valid sentences and proofs.

So it's not that the foundations of math are self-referential, it's just that they can't be "formally defined all the way down": our capacity for logical reasoning must exist to some degree before we can describe it, because if it didn't, there'd be no tools to define it into existence. Propositional and predicate logic are the levels where we start being formal, so you should consider the exposition of these systems (at least in the context of introducing set theory) to be informal or self-evident.

When we eventually want to study logical systems abstractly and need a way to express their properties as mathematical statements (i.e. do metamathematics), we can use set theory to define objects that model the logical primitives and then prove things about those objects. I'd argue that these objects are not actually "at work" when we use logic; they're just mathematical models for a real-world activity.

How are the axioms defined in the fundamentals of set theory

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