Definition:Class (Class Theory)/Zermelo-Fraenkel

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Definition

Denote with $\textrm{ZF}$ the language of set theory endowed with the Zermelo-Fraenkel axioms.

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.

This definition "overloads" the membership symbol $\in$ since its operands could now be either sets or classes.

That such does not lead to ambiguity is proved on Class Membership Extension of Set Membership.

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Class Variables

In deriving general results about $\textrm{ZF}$ which mention classes, it is often convenient to have class variables, which denote an arbitrary class.

By convention, these variables are taken on $\mathsf{Pr} \infty \mathsf{fWiki}$ to be the (start of) the capital Latin alphabet, i.e. $A, B, C$, and so on.

Caution

Unlike in von Neumann-Bernays-Gödel set theory, it is prohibited to quantify over classes.

That is, expressions like:

$\forall A: \map P A$

are ill-defined; admitting them without further consideration would lead us to Russell's Paradox.

Sources

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