Transitive set

Transitive set

In set theory, a set (or class) "A" is transitive, if
* whenever "x" ∈ "A", and "y" ∈ "x", then "y" ∈ "A", or, equivalently,
* whenever "x" ∈ "A", and "x" is not an urelement, then "x" is a subset of "A".

The transitive closure of a set "A" is the smallest (with respect to inclusion) transitive set "B" which contains "A". Suppose one is given a set "X", then the transitive closure of "X" is::igcup { X, igcup X, igcup igcup X, igcup igcup igcup X, igcup igcup igcup igcup X, ... } .

Transitive classes are often used for construction of interpretations of set theory in itself, usually called inner models. The reason is that properties defined by bounded formulas are absolute for transitive classes.

An ordinal number may be defined as an hereditarily transitive set, that is, a transitive set whose members are also transitive.

A set "X" is transitive if and only if igcup X subseteq X.

A set "X" which does not contain urelements is transitive if and only if X subset mathcal{P}(X).

Transitive classes

Similarly, a class "M" is transitive if every element of "M" is a subset of "M".

Transitive models of set theory

A transitive set (or class) that is a model of a formal system of set theory is called a transitive model of the system. Transitivity is an important factor in determining the absoluteness of formulas.

ee also

*Union (set theory)
*Power set


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