Def/Class

Logical Connections

This definition logically relies on the following definitions and statements: Def/Unary predicate, Def/Russell predicate

The following statements and definitions logically rely on the material of this page: Def/Multiplication of cardinal numbers

To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Logic and foundations

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 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.

This article has been written by: User:Marty