Transitive set
in set theory, a set whose elements are all subsets
In set theory, a branch of mathematics, a set
A
{\displaystyle A}
is called transitive if either of the following equivalent conditions holds:
whenever
x
∈
A
{\displaystyle x\in A}
, and
y
∈
x
{\displaystyle y\in x}
, then
y
∈
A
{\displaystyle y\in A}
.
whenever
x
∈
A
{\displaystyle x\in A}
, and
x
{\displaystyle x}
is not an urelement, then
x
{\displaystyle x}
is a subset of
A
{\displaystyle A}
.
Similarly, a class
M
{\displaystyle M}
is transitive if every element of
M
{\displaystyle M}
is a subset of
M
{\displaystyle M}
.
Examples
Using the definition of ordinal numbers suggested by John von Neumann, ordinal numbers are defined as hereditarily transitive sets: an ordinal number is a transitive set whose members are also transitive (and thus ordinals). The class of all ordinals is a transitive class.
Any of the stages
V
α
{\displaystyle V_{\alpha }}
and
L
α
{\displaystyle L_{\alpha }}
leading to the construction of the von Neumann universe
V
{\displaystyle V}
and Gödel's constructible universe
L
{\displaystyle L}
are transitive sets. The universes
V
{\displaystyle V}
and
L
{\displaystyle L}
themselves are transitive classes.
This is a complete list of all finite transitive sets with up to 20 pairs of brackets:
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Properties
A set
X
{\displaystyle X}
is transitive if and only if
⋃
X
⊆
X
{\displaystyle \textstyle \bigcup X\subseteq X}
, where
⋃
X
{\displaystyle \textstyle \bigcup X}
is the union of all elements of
X
{\displaystyle X}
that are sets; formally,
⋃
X
=
{
y
∣
∃
x
∈
X
:
y
∈
x
}
{\displaystyle \bigcup X=\{y\mid \exists x\in X:y\in x\}}
.
Moreover, if
X
{\displaystyle X}
is transitive, then
⋃
X
{\displaystyle \textstyle \bigcup X}
is transitive.
Begin with the source’s own compact description: “Transitive set” is in set theory, a set whose elements are all subsets. The dossier treats that line as a proposition to test through Transitive, theory and whose, not as a finished interpretation.
Why this record matters
The phrase “in set theory, a set whose elements are all subsets” supplies a clear boundary for inquiry. It also exposes the unanswered questions: who defined that boundary, when it became stable and which sources sit outside it.
The citation trail is more important than the brevity of the summary: it shows where individual claims can be examined in context. The source revision retrieved here is dated Aug 23, 2026. The linked authority identifier is Q671944. None of the 0 selected statements returned an explicit reference.
The absence of detail may reflect summary conventions rather than a lack of surviving documentation. The lead is largely declarative, so disagreement and counter-evidence require a deliberate search beyond the opening account. Authority statements aid reconciliation but still require their own references, qualifiers and ranks to be checked.
How to read it
Use the entry as an orientation point, then follow its citations and revision history. Names, dates and institutional relationships should be checked against the original record.
- Subject orientation
- Search vocabulary
- Locating named sources
The closest primary source, responsible institution and strongest cited specialist reference.
Three-step research path
- Establish the record: confirm the title “Transitive set”, its source revision and the description used here.
- Expand the search: follow Transitive set primary sources, Transitive set archive and Transitive research across catalogues and specialist indexes.
- Test the account: compare the strongest cited source with the responsible institution’s current record and note any disagreement.
Questions for further research
- Which source most directly establishes the central claim about “Transitive set”?
- Which institution is responsible for the underlying evidence?
- Which cited source is closest to the event, object or claim?
Search terms from this dossier
This entry incorporates text from “Transitive set” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.