Material set theory

Axioms of Permutation for classes

The Axioms of Permutation are a pair of possible axioms for Material set theory with classes¹: nbg

(∀ℭ𝔩𝔰⁡𝑋)(∃ℭ𝔩𝔰⁡𝑍)(∀𝑢)(∀𝑣)(∀𝑤)[(𝑢,𝑣,𝑤)∈𝑍⟺(𝑣,𝑤,𝑢)∈𝑋](∀ℭ𝔩𝔰⁡𝑋)(∃ℭ𝔩𝔰⁡𝑍)(∀𝑢)(∀𝑣)(∀𝑤)[(𝑢,𝑣,𝑤)∈𝑍⟺(𝑢,𝑤,𝑣)∈𝑋]

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.

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Footnotes

1. 2015. Introduction to Mathematical Logic, §4.1, p. 237, B6–7 ↩