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 .