Stolen base percentage

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

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

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

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