set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

, a supertransitive class is a transitive class which includes as a subset the

power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...

of each of its elements. Formally, let ''A'' be a transitive class. Then ''A'' is supertransitive if and only if :

(\forall x)(x\in A \to \mathcal(x) \subseteq A).

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

References

{{reflist Set theory

See also

References