End extension

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

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

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

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