In
set theory
Set theory is the branch of mathematical logic that studies 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 mathematics, is mostly conce ...
, a supertransitive class is a
transitive class
In set theory, a branch of mathematics, a set A is called transitive if either of the following equivalent conditions hold:
* whenever x \in A, and y \in x, then y \in A.
* whenever x \in A, and x is not an urelement, then x is a subset of A.
Sim ...
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
:
Here ''P''(''x'') denotes the power set of ''x''.
[''P''(''x'') must be a set by ]axiom of power set
In mathematics, the axiom of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory.
In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:
:\forall x \, \exists y \, \forall z \, \in y \iff \forall w \ ...
, since each element ''x'' of a class ''A'' must be a set (Theorem 4.6 in Takeuti's text above).
See also
*
Transitive set
In set theory, a branch of mathematics, a set A is called transitive if either of the following equivalent conditions hold:
* whenever x \in A, and y \in x, then y \in A.
* whenever x \in A, and x is not an urelement, then x is a subset of A.
Simil ...
*
Rank (set theory)
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by ''V'', is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory (ZF ...
References
{{reflist
Set theory