End extension

In model theory and set theory, which are disciplines within mathematics, a model ${\displaystyle {\mathfrak {B}}=\langle B,F\rangle }$ of some axiom system of set theory ${\displaystyle T}$ in the language of set theory is an end extension of ${\displaystyle {\mathfrak {A}}=\langle A,E\rangle }$, in symbols ${\displaystyle {\mathfrak {A}}\subseteq _{\text{end}}{\mathfrak {B}}}$, if

1. ${\displaystyle {\mathfrak {A}}}$ is a substructure of ${\displaystyle {\mathfrak {B}}}$, (i.e., ${\displaystyle A\subseteq B}$ and ${\displaystyle E=F|_{A}}$), and
2. ${\displaystyle b\in A}$ whenever ${\displaystyle a\in A}$ and ${\displaystyle bFa}$ hold, i.e., no new elements are added by ${\displaystyle {\mathfrak {B}}}$ to the elements of ${\displaystyle A}$.¹

The second condition can be equivalently written as ${\displaystyle \{b\in A:bEa\}=\{b\in B:bFa\}}$ for all ${\displaystyle a\in A}$.

For example, ${\displaystyle \langle B,\in \rangle }$ is an end extension of ${\displaystyle \langle A,\in \rangle }$ if ${\displaystyle A}$ and ${\displaystyle B}$ are transitive sets, and ${\displaystyle A\subseteq B}$.

A related concept is that of a top extension (also known as rank extension), where a model ${\displaystyle {\mathfrak {B}}=\langle B,F\rangle }$ is a top extension of a model ${\displaystyle {\mathfrak {A}}=\langle A,E\rangle }$ if ${\displaystyle {\mathfrak {A}}\subseteq _{\text{end}}{\mathfrak {B}}}$ and for all ${\displaystyle a\in A}$ and ${\displaystyle b\in B\setminus A}$, we have ${\displaystyle rank(b)>rank(a)}$, where ${\displaystyle rank(\cdot )}$ denotes the rank of a set.

Keisler and Morley showed that every countable model of ZF has an end extension which is also an elementary extension.² If the elementarity requirement is weakened to being elementary for formulae that are ${\displaystyle \Sigma _{n}}$ on the Lévy hierarchy, every countable structure in which ${\displaystyle \Sigma _{n}}$-collection holds has a ${\displaystyle \Sigma _{n}}$-elementary end extension.³