# End extension

In model theory and set theory, which are disciplines within mathematics, a model
of some axiom system of set theory
in the language of set theory is an **end extension** of
, in symbols
, if

- is a substructure of , and
- whenever and hold, i.e., no new elements are added by to the elements of .

The following is an equivalent definition of end extension: is a substructure of , and for all .

For example, is an end extension of if and are transitive sets, and .

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