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 $\mathfrak{A}$.

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