[[Material set theory]]
# Axioms of Permutation for classes

The **Axioms of Permutation** are a pair of possible axioms for [[Material set theory]] with [[Class|classes]][^2015]: #m/def/set/nbg 
$$
\begin{align*}
(\forall \chood X)(\exists \chood Z)(\forall u)(\forall v)(\forall w)[(u,v,w) \in Z \iff (v,w,u) \in X] \\
(\forall \chood X)(\exists \chood Z)(\forall u)(\forall v)(\forall w)[(u,v,w) \in Z \iff (u,w,v) \in X]
\end{align*}
$$
Sometimes, the first is called the **Axiom of Circular Permutation** and the second is called the **Axiom of Transposition**.
These help ensure that any reordering of pairs in some class of ordered pairs results in a class.

  [^2015]: 2015\. [[Sources/@mendelsonIntroductionMathematicalLogic2015|Introduction to Mathematical Logic]], §4.1, p. 237, B6–7


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