Definition:Class (Class Theory)

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Definition

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


In class builder notation, this is written as:

$\left\{{x : p \left({x}\right)}\right\}$

where $p \left({x}\right)$ is a statement containing $x$ as a free variable.

This is read:

All $x$ such that $p \left({x}\right)$ 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:

$\left\{{x : P \left({x}\right)}\right\}$


where $P \left({x}\right)$ is a statement containing $x$ as a free variable.


More formally, a class $\left\{ {x : P \left({x}\right)}\right\}$ serves to define the following definitional abbreviations involving the membership symbol:

\(\displaystyle y \in \left\{ {x: P \left({x}\right)}\right\}\) \(\quad \text{for} \quad\) \(\displaystyle P \left({y}\right)\)
\(\displaystyle \left\{ {x: P \left({x}\right)}\right\} \in y\) \(\quad \text{for} \quad\) \(\displaystyle \exists z \in y: \forall x: \left({x \in z \iff P \left({x}\right)}\right)\)
\(\displaystyle \left\{ {x: P \left({x}\right)}\right\} \in \left\{ {y: Q \left({y}\right)}\right\}\) \(\quad \text{for} \quad\) \(\displaystyle \exists z: \left({Q \left({z}\right) \land \forall x: \left({x \in z \iff P \left({x}\right)}\right)}\right)\)

where:

$x, y ,z$ are variables of $\textrm{ZF}$
$P, Q$ are propositional functions.

Through these "rules", every statement involving $\left\{{x : P \left({x}\right) }\right\}$ 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 iff:

$\neg \exists x: x = A$

where $x$ is a set.


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


Also see

  • Results about classes can be found here.


Sources