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Supertransitive class

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

[2]

Here P(x) denotes the power set of x.[3]

See also

[edit]

References

[edit]
  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).
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Supertransitive class
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