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
  1. is a substructure of,, and
  2. 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, where a model is a top extension of a model if and for all and, we have, where denotes the rank of a set.

Existence

Keisler and Morley showed that every countable model of ZF has an end extension which is also an elementary extension. If the elementarity requirement is weakened to being elementary for formulae that are on the Lévy hierarchy, every countable structure in which -collection holds has a -elementary end extension.