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Set membership is neither defined nor specified in the signature of ZFC. Apart from equality, there is another binary relation in the signature of ZFC, which we interpret as set membership.

Edit: Andrej Bauer has convinced me that set membership is specified in the signature of ZFC. Also see my comment below. However, this is just terminology. What follows should be more important:

The axioms of ZFC describe the properties of set membership. The unary predicates that you talk about are not a priori set, but what we call classes. It happens that certain classes coincide with certain sets, namely when the members of the set are precisely the sets that satisfy the predicate.

In the language of set theory you can quantify over set, as these are the objects that this language talks about, but not over classes, the unary predicates.

The usual reason for confusion that arises in the context of set theory is the fact that when doing math, we already use the relation $\in$ to talk about the things that we are studying. When talking about a structure for a first order language, we talk about a set with certain relations, functions, and constants. When considering a model of ZFC we need to be really careful: There are the things that we consider as sets in the "real world", i.e., the universe of sets in which we do mathematics (and which we assume to satisfy ZFC), and then there is a model of ZFC, let's call it $M$, which is itself a set in the universe of sets and which carries a binary relation $E$ such that $(M,E)$ satisfies ZFC.

Now, from the perspective of $M$, the elements of $M$ are the sets, and they are related to each other by $E$. $E$ does not have to be the real $\in$, even though it can be. The classes of $M$ are subsets of $M$ in the "universe of all sets"-way, but they are not known to $M$. There are some subsets $A$ of $M$ such that there is $a\in M$ with $\{b\in M:b E a\}=A$. If $A$ is a class of $M$, i.e., the collection of elements of $M$ satisfying a certain unary predicate, we identify this class with the "set" $a$ (set in the sense of $M$). But not all classes of $M$ can be identified with sets of $M$ in this way, for example all of $M$ or the class of all ordinals of $M$.

I hope this helps.

The axioms of ZFC describe the properties of set membership. The unary predicates that you talk about are not a priori set, but what we call classes. It happens that certain classes coincide with certain sets, namely when the members of the set are precisely the sets that satisfy the predicate.

In the language of set theory you can quantify over set, as these are the objects that this language talks about, but not over classes, the unary predicates.

The usual reason for confusion that arises in the context of set theory is the fact that when doing math, we already use the relation $\in$ to talk about the things that we are studying. When talking about a structure for a first order language, we talk about a set with certain relations, functions, and constants. When considering a model of ZFC we need to be really careful: There are the things that we consider as sets in the "real world", i.e., the universe of sets in which we do mathematics (and which we assume to satisfy ZFC), and then there is a model of ZFC, let's call it $M$, which is itself a set in the universe of sets and which carries a binary relation $E$ such that $(M,E)$ satisfies ZFC.

Now, from the perspective of $M$, the elements of $M$ are the sets, and they are related to each other by $E$. $E$ does not have to be the real $\in$, even though it can be. The classes of $M$ are subsets of $M$ in the "universe of all sets"-way, but they are not known to $M$. There are some subsets $A$ of $M$ such that there is $a\in M$ with $\{b\in M:b E a\}=A$. If $A$ is a class of $M$, i.e., the collection of elements of $M$ satisfying a certain unary predicate, we identify this class with the "set" $a$ (set in the sense of $M$). But not all classes of $M$ can be identified with sets of $M$ in this way, for example all of $M$ or the class of all ordinals of $M$.

I hope this helps.

Set membership is neither defined nor specified in the signature of ZFC. Apart from equality, there is another binary relation in the signature of ZFC, which we interpret as set membership.

The axioms of ZFC say (or specify, if you wish) the properties of set membership. The unary predicates that you talk about are not a priori set, but what we call classes. It happens that certain classes coincide with certain sets, namely when the members of the set are precisely the sets that satisfy the predicate.

In the language of set theory you can quantify over set, as these are the objects that this language talks about, but not over classes, the unary predicates.

The usual reason for confusion that arises in the context of set theory is the fact that when doing math, we already use the relation $\in$ to talk about the things that we are studying. When talking about a structure for a first order language, we talk about a set with certain relations, functions, and constants. When considering a model of ZFC we need to be really careful: There are the things that we consider as sets in the "real world", i.e., the universe of sets in which we do mathematics (and which we assume to satisfy ZFC), and then there is a model of ZFC, let's call it $M$, which is itself a set in the universe of sets and which carries a binary relation $E$ such that $(M,E)$ satisfies ZFC.

Now, from the perspective of $M$, the elements of $M$ are the sets, and they are related to each other by $E$. $E$ does not have to be the real $\in$, even though it can be. The classes of $M$ are subsets of $M$ in the "universe of all sets"-way, but they are not known to $M$. There are some subsets $A$ of $M$ such that there is $a\in M$ with $\{b\in M:b E a\}=A$. If $A$ is a class of $M$, i.e., the collection of elements of $M$ satisfying a certain unary predicate, we identify this class with the "set" $a$ (set in the sense of $M$). But not all classes of $M$ can be identified with sets of $M$ in this way, for example all of $M$ or the class of all ordinals of $M$.

I hope this helps.

Set membership is neither defined nor specified in the signature of ZFC. Apart from equality, there is another binary relation in the signature of ZFC, which we interpret as set membership.

Edit: Andrej Bauer has convinced me that set membership is specified in the signature of ZFC. Also see my comment below. However, this is just terminology. What follows should be more important:

The axioms of ZFC describe the properties of set membership. The unary predicates that you talk about are not a priori set, but what we call classes. It happens that certain classes coincide with certain sets, namely when the members of the set are precisely the sets that satisfy the predicate.

In the language of set theory you can quantify over set, as these are the objects that this language talks about, but not over classes, the unary predicates.

The usual reason for confusion that arises in the context of set theory is the fact that when doing math, we already use the relation $\in$ to talk about the things that we are studying. When talking about a structure for a first order language, we talk about a set with certain relations, functions, and constants. When considering a model of ZFC we need to be really careful: There are the things that we consider as sets in the "real world", i.e., the universe of sets in which we do mathematics (and which we assume to satisfy ZFC), and then there is a model of ZFC, let's call it $M$, which is itself a set in the universe of sets and which carries a binary relation $E$ such that $(M,E)$ satisfies ZFC.

Now, from the perspective of $M$, the elements of $M$ are the sets, and they are related to each other by $E$. $E$ does not have to be the real $\in$, even though it can be. The classes of $M$ are subsets of $M$ in the "universe of all sets"-way, but they are not known to $M$. There are some subsets $A$ of $M$ such that there is $a\in M$ with $\{b\in M:b E a\}=A$. If $A$ is a class of $M$, i.e., the collection of elements of $M$ satisfying a certain unary predicate, we identify this class with the "set" $a$ (set in the sense of $M$). But not all classes of $M$ can be identified with sets of $M$ in this way, for example all of $M$ or the class of all ordinals of $M$.

I hope this helps.

Set membership is neither defined nor specified in the signature of ZFC. Apart from equality, there is another binary relation in the signature of ZFC, which we interpret as set membership.

The axioms of ZFC say (or specify, if you wish) the properties of set membership. The unary predicates that you talk about are not a priori set, but what we cal classes. It happens that certain classes coincide with certain sets, namely when the members of the set are precisely the sets that satisfy the predicate.

In the language of set theory you can quantify over set, as these are the objects that this language talks about, but not over classes, the unary predicates.

I hope this helps.

Set membership is neither defined nor specified in the signature of ZFC. Apart from equality, there is another binary relation in the signature of ZFC, which we interpret as set membership.

The axioms of ZFC say (or specify, if you wish) the properties of set membership. The unary predicates that you talk about are not a priori set, but what we call classes. It happens that certain classes coincide with certain sets, namely when the members of the set are precisely the sets that satisfy the predicate.

In the language of set theory you can quantify over set, as these are the objects that this language talks about, but not over classes, the unary predicates.

The usual reason for confusion that arises in the context of set theory is the fact that when doing math, we already use the relation $\in$ to talk about the things that we are studying. When talking about a structure for a first order language, we talk about a set with certain relations, functions, and constants. When considering a model of ZFC we need to be really careful: There are the things that we consider as sets in the "real world", i.e., the universe of sets in which we do mathematics (and which we assume to satisfy ZFC), and then there is a model of ZFC, let's call it $M$, which is itself a set in the universe of sets and which carries a binary relation $E$ such that $(M,E)$ satisfies ZFC.

Now, from the perspective of $M$, the elements of $M$ are the sets, and they are related to each other by $E$. $E$ does not have to be the real $\in$, even though it can be. The classes of $M$ are subsets of $M$ in the "universe of all sets"-way, but they are not known to $M$. There are some subsets $A$ of $M$ such that there is $a\in M$ with $\{b\in M:b E a\}=A$. If $A$ is a class of $M$, i.e., the collection of elements of $M$ satisfying a certain unary predicate, we identify this class with the "set" $a$ (set in the sense of $M$). But not all classes of $M$ can be identified with sets of $M$ in this way, for example all of $M$ or the class of all ordinals of $M$.

I hope this helps.