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In Wikipedia, it says that ZFC is a one-sorted theory in first-order logic. However, I was not really able to comprehend the later parts that seem to elaborate on that point. Can anyone explain the relationship between ZFC and first-order logic?

Thanks.

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    $\begingroup$ Judging by your questions I'd say that you try to understand how set theory (and ZFC in particular) can be used to serve as a foundational framework to mathematics. I'd suggest first to study some basic set theory, some predicate calculus and gain better understanding what is a language, what are structures; some model theory should come after that. Then it can be easier to understand how set theory is a first order theory, but can encapsulate high-order logics within a model of ZFC, thus allowing the development of mathematics inside set theory. $\endgroup$
    – Asaf Karagila
    Feb 12, 2012 at 17:53

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First-order logic is a framework in which one can only quantify over elements of the universe. In second-order logic one is allowed to quantify over subsets of the universe.

For example, the axiom schema of induction in Peano axioms is a schema in which for every formula we add a first-order axiom talk about this formula. However if one allows second-order axioms the schema shrinks into one axiom, namely "If $A$ is an inductive set, then $A=\mathbb N$".

In the first formulation for every formula which can be used to define a set we say that if the set it defines has inductive properties then it is indeed everything; on the second-order axiom we are allowed to quantify over subsets of the universe so we say that every subset which has this property has got to be equal to $\mathbb N$.

Many-sorted logic is a logic in which we distinct between several types of elements in our universe. For example set theory with atoms, which allows non-set objects in the universe. In this language there are predicates to indicate whether an object is a set or an atom. It cannot be both.

Second-order theories can be made into first-order theory by expanding the universe to include sets, and adding a distinction between the elements of the original universe and sets. Now quantification over "every set such that ..." is again first-order since those sets are elements of the universe.

ZFC is a first-order logic theory, it allows only to quantify over elements of the universe. It is also one-sorted since there is only one type of elements in a universe of ZFC, namely sets.

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  • $\begingroup$ Since in ZFC some subsets of the universe are in fact elements of the universe (aka sets), would it make sense to say something informal like the following to illustrate the subtlety? In ZFC we can quantify over $\mathcal{P}(x)$ for any set $x \in \mathbb{V}$ but we cannot quantify over the entire "$\mathcal{P}(\mathbb{V})$" (whatever that means). $\endgroup$
    – Pellenthor
    Apr 7, 2020 at 9:50
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    $\begingroup$ Well, the notation $\mathcal P$ denotes "all subsets" which are sets, by definition, so the notation is usually indicating that $\mathcal P(V)=V$. But if you mean externally, then yes. In that case, $\mathcal P(V)$ would be all subcollections of the universe. The sets, the classes, the undefinable collections. All of them. $\endgroup$
    – Asaf Karagila
    Apr 7, 2020 at 10:30
  • $\begingroup$ Yes I did mean "externally" (which is why I put it in quotes) but you did make a very important point. Thanks. $\endgroup$
    – Pellenthor
    Apr 8, 2020 at 12:46
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ZFC is a first-order theory over a language with two binary predicates, $=$ and $\in$. It is given by a bunch of axioms, which are sentences in first-order logic.

Second-order logic is many-sorted: it has two different types of variables, basic variables and predicates. Contrastingly, the axioms of ZFC only talk about one kind of object, sets.

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    $\begingroup$ There are also many-sorted first-order theories. $\endgroup$
    – Zhen Lin
    Feb 12, 2012 at 11:33

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