[[Material set theory]]
# Axiom of Purity

The **Axiom of Purity** is a possible axiom of [[material set theory]] included in most axiomatizations of [[ZF]]: #m/def/set
$$
\begin{align*}
(\forall x)[\shood(x)]
\end{align*}
$$
which is to say, the domain of discourse is restricted to sets,
and thus every set is a [[pure set]] and there are no [[Urelement|urelements]].
Usually, purity is not taken as an axiom and instead everything in the universe is implicitly taken to be a set, doing away with the sethood predicate $\shood$.
Since many axioms of material set theory must be modified to allow for urelements,
these notes will not assume purity instead opting to treat it as a separate, optional axiom.

#
---
#state/tidy | #lang/en | #SemBr