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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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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. –  Asaf Karagila Feb 12 '12 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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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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There are also many-sorted first-order theories. –  Zhen Lin Feb 12 '12 at 11:33

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