Definition:Class (Class Theory)

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This page is about Class in the context of Class Theory. For other uses, see Class.

Definition

A class is a collection of all sets such that a particular condition holds.


In class-builder notation, this is written as:

$\set {x: \map p x}$

where $\map p x$ is a statement containing $x$ as a free variable.

This is read:

All $x$ such that $\map p x$ holds.


Definition in ZF Set Theory

A class in $\textrm{ZF}$ is a formal vehicle capturing the intuitive notion of a class, namely a collection of all sets such that a particular condition $P$ holds.

In $\textrm{ZF}$, classes are written using class-builder notation:

$\set {x : \map P x}$

where $\map P x$ is a well-formed formula containing $x$ as a free variable.


More formally, a class $\set {x: \map P x}$ serves to define the following definitional abbreviations involving the membership symbol:

\(\ds y \in \set {x: \map P x}\) \(\text{for}\) \(\ds \map P y\)
\(\ds \set {x: \map P x} \in y\) \(\text{for}\) \(\ds \exists z \in y: \forall x: \paren {x \in z \iff \map P x}\)
\(\ds \set {x: \map P x} \in \set {y: \map Q y}\) \(\text{for}\) \(\ds \exists z: \paren {\map Q z \land \forall x: \paren {x \in z \iff \map P x} }\)

where:

$x, y ,z$ are variables of $\textrm{ZF}$
$P, Q$ are well-formed formulas.

Through these "rules", every statement involving $\set {x: \map P x}$ can be reduced to a simpler statement involving only the basic language of set theory.


Proper Class

A proper class is a class which is not a set.

That is, $A$ is a proper class if and only if:

$\neg \exists x: x = A$

where $x$ is a set.


Also see

  • Results about class theory can be found here.


Sources

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