Definition 1. If $\varphi(x, p)$ is a formula with free variables among $x$ and $p$, we define the class
\[C = \{x : \varphi(x, p)\}\]as an object such that
\[\forall x :\enspace x \in C \enspace\lrimp\enspace \varphi(x, p).\]In $\ZF$ and similar set theories, every mathematical object is a set. Nevertheless, we introduce classes, an informal object, through which we can talk about formulas in the same way we talk about sets.
It is important to remember that classes are only syntactic sugar. Arguments involving classes are ultimately reduced to formulas.
Definition 2. Relations and operations on classes, such as
\[C = D ,\enspace% C \subseteq D ,\enspace% C \cup D ,\enspace% C \cap D ,\enspace% C \setdif D ,\enspace% \bigcup C ,\enspace% \bigcap C ,\]are defined similarly to those on sets. For example, for any two classes $C$ and $D$, $C \subseteq D$ ($C$ is a subclass of $D$) if
\[\forall x :\enspace x \in C \enspace\rimp\enspace x \in D.\]Definition 3. A class $C$ is a set if there is a set $X$ such that
\[\forall x :\enspace x \in C \enspace\lrimp\enspace x \in X.\]A class that is not a set is called a proper class.
References
- [1]
- Wikipedia contributors. Class (set theory). Wikipedia, The Free Encyclopedia.
- [2]
- Thomas Jech. Set Theory. 3rd Millennium ed, rev. and expanded. Springer, 2002. pp. 5–6.