Category:Zermelo-Fraenkel Class Theory

From ProofWiki
Jump to: navigation, search

This category contains results about classes in ZF set theory.
Definitions specific to this category can be found in Definitions/Zermelo-Fraenkel Class 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:

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


$x, y ,z$ 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.