Class of mathematical set whose elements are all subsets
In set theory , a branch of mathematics , a set
A
{\displaystyle A}
transitive if either of the following equivalent conditions hold:
whenever
x
∈
A
{\displaystyle x\in A}
y
∈
x
{\displaystyle y\in x}
y
∈
A
{\displaystyle y\in A}
whenever
x
∈
A
{\displaystyle x\in A}
x
{\displaystyle x}
urelement , then
x
{\displaystyle x}
subset of
A
{\displaystyle A}
Similarly, a class
M
{\displaystyle M}
M
{\displaystyle M}
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 }}
L
α
{\displaystyle L_{\alpha }}
von Neumann universe
V
{\displaystyle V}
Gödel's constructible universe
L
{\displaystyle L}
universes
V
{\displaystyle V}
L
{\displaystyle L}
This is a complete list of all finite transitive sets with up to 20 brackets:[1]
{
}
,
{\displaystyle \{\},}
{
{
}
}
,
{\displaystyle \{\{\}\},}
{
{
}
,
{
{
}
}
}
,
{\displaystyle \{\{\},\{\{\}\}\},}
{
{
}
,
{
{
}
}
,
{
{
{
}
}
}
}
,
{\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\}\},}
{
{
}
,
{
{
}
}
,
{
{
}
,
{
{
}
}
}
}
,
{\displaystyle \{\{\},\{\{\}\},\{\{\},\{\{\}\}\}\},}
{
{
}
,
{
{
}
}
,
{
{
{
}
}
}
,
{
{
{
{
}
}
}
}
}
,
{\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\{\{\}\}\}\}\},}
{
{
}
,
{
{
}
}
,
{
{
{
}
}
}
,
{
{
}
,
{
{
}
}
}
}
,
{\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\},\{\{\}\}\}\},}
{
{
}
,
{
{
}
}
,
{
{
{
}
}
}
,
{
{
}
,
{
{
{
}
}
}
}
}
,
{\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\},\{\{\{\}\}\}\}\},}
{
{
}
,
{
{
}
}
,
{
{
{
}
}
}
,
{
{
{
}
}
,
{
{
{
}
}
}
}
}
,
{\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\{\}\},\{\{\{\}\}\}\}\},}
{
{
}
,
{
{
}
}
,
{
{
{
}
,
{
{
}
}
}
}
,
{
{
}
,
{
{
}
}
}
}
,
{\displaystyle \{\{\},\{\{\}\},\{\{\{\},\{\{\}\}\}\},\{\{\},\{\{\}\}\}\},}
{
{
}
,
{
{
}
}
,
{
{
{
}
}
}
,
{
{
}
,
{
{
}
}
,
{
{
{
}
}
}
}
}
,
{\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\},\{\{\}\},\{\{\{\}\}\}\}\},}
{
{
}
,
{
{
}
}
,
{
{
}
,
{
{
}
}
}
,
{
{
}
,
{
{
}
,
{
{
}
}
}
}
}
,
{\displaystyle \{\{\},\{\{\}\},\{\{\},\{\{\}\}\},\{\{\},\{\{\},\{\{\}\}\}\}\},}
{
{
}
,
{
{
}
}
,
{
{
}
,
{
{
}
}
}
,
{
{
{
}
}
,
{
{
}
,
{
{
}
}
}
}
}
,
{\displaystyle \{\{\},\{\{\}\},\{\{\},\{\{\}\}\},\{\{\{\}\},\{\{\},\{\{\}\}\}\}\},}
{
{
}
,
{
{
}
}
,
{
{
{
}
}
}
,
{
{
{
{
}
}
}
}
,
{
{
}
,
{
{
}
}
}
}
,
{\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\{\{\}\}\}\},\{\{\},\{\{\}\}\}\},}
{
{
}
,
{
{
}
}
,
{
{
}
,
{
{
}
}
}
,
{
{
}
,
{
{
}
}
,
{
{
}
,
{
{
}
}
}
}
}
,
{\displaystyle \{\{\},\{\{\}\},\{\{\},\{\{\}\}\},\{\{\},\{\{\}\},\{\{\},\{\{\}\}\}\}\},}
{
{
}
,
{
{
}
}
,
{
{
{
}
}
}
,
{
{
{
{
}
}
}
}
,
{
{
{
{
{
}
}
}
}
}
}
,
{\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\{\{\}\}\}\},\{\{\{\{\{\}\}\}\}\}\},}
{
{
}
,
{
{
}
}
,
{
{
{
}
}
}
,
{
{
{
{
}
}
}
}
,
{
{
}
,
{
{
{
}
}
}
}
}
,
{\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\{\{\}\}\}\},\{\{\},\{\{\{\}\}\}\}\},}
{
{
}
,
{
Zdroj: https://en.wikipedia.org?pojem=Transitive_set
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