Supertransitive class
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Short description: Transitive class including powersets of elements
In set theory, a supertransitive class is a transitive class[1] 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
- [math]\displaystyle{ (\forall x)(x\in A \to \mathcal{P}(x) \subseteq A). }[/math][2]
Here P(x) denotes the power set of x.[3]
See also
- Rank (set theory)
References
- ↑ 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.
- ↑ 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.
- ↑ 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).
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