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In set theory, a supertransitive class is a transitive class which includes as a subset the power set of each of its elements.

Formally, let A be a transitive class. Then A is supertransitive if and only if

${\displaystyle (\forall x)(x\in A\to {\mathcal {P}}(x)\subseteq A).}$

Here P(x) denotes the power set of x.

See also

• Rank (set theory)

References

1. Any element of a transitive set must also be its subset. See Definition 7.1 of Zaring W.M., G. Takeuti (1971). Introduction to axiomatic set theory (2nd, rev. ed.). New York: Springer-Verlag. ISBN 0387900241.
2. See Definition 9.8 of Zaring W.M., G. Takeuti (1971). Introduction to axiomatic set theory (2nd, rev. ed.). New York: Springer-Verlag. ISBN 0387900241.
3. P(x) must be a set by axiom of power set, since each element x of a class A must be a set (Theorem 4.6 in Takeuti's text above).

• Set theory