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

• ${\displaystyle {\mathfrak {A}}}$ is a substructure of ${\displaystyle {\mathfrak {B}}}$, and
• ${\displaystyle b\in A}$ whenever ${\displaystyle a\in A}$ and ${\displaystyle bFa}$ hold, i.e., no new elements are added by ${\displaystyle {\mathfrak {B}}}$ to the elements of ${\displaystyle A}$.

The following is an equivalent definition of end extension: ${\displaystyle {\mathfrak {A}}}$ is a substructure of ${\displaystyle {\mathfrak {B}}}$, and ${\displaystyle \{b\in A:bEa\}=\{b\in B:bFa\}}$ 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 ${\displaystyle A\subseteq B}$.