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 , (i.e., and ), and
- whenever and hold, i.e., no new elements are added by to the elements of .
The second condition can be equivalently written as for all .
For example, is an end extension of if and are transitive sets, and .
A related concept is that of a top extension (also known as rank extension), where a model is a top extension of a model if and for all and , we have , where denotes the rank of a set.