Def/Class

 =Sets and Classes=

Membership of sets in classes
Note that $\{ x \vert \Phi(x) \}$ may not be a set. However, membership of sets in classes can be defined as follows: if $y$ is a set, then we say that $y \in \{ x \vert \Phi(x) \}$ if $\Phi(y)$ is true.

Membership of classes in other classes is not defined.

Equality of classes
Suppose that $\Phi(x)$ and $\Psi(x)$ are two unary predicates. Then, we say that there is an equality of classes: $$\{ x \vert \Phi(x) \} = \{ x \vert \Psi(x) \},$$ if the sentence $\forall x, \Phi(x) \Leftrightarrow \Psi(x)$ is true.

A class might be a set
If $y$ is a set, then we say that $y = \{x \vert \Phi(x) \}$, if the following sentence is true: $$\forall a (a \in y \Leftrightarrow \Phi(a) ).$$ In this case, we say that the class $\{x \vert \Phi(x) \}$ is represented by $y$.

In this case, we say that the class $\{ x \vert \Phi(x) \}$ is a set. Otherwise, if $\{x \vert \Phi(x) \}$ is not a set, then $\{x \vert \Phi(x) \}$ is called a defines::proper class. }}

A class might not be a set
When $\Phi(x)$ is the Russell predicate, the class $\{ x \vert \Phi(x) \} = \{ x \vert x \not \in x \}$ is not a set.

A more simple example is the "class of all sets", given for example by the predicate $x = x$. The class $\{x \vert x = x \}$ is not a set.

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