= Solution
The formula saying that the $E$-class of $x$ has exactly $n$ elements uses quantifiers and distinguishes elements in differently sized classes. No quantifier-free one-variable formula in the language $\{E\}$ can do so, since its only atomic information is $x=x$ and $E(x,x)$. Thus $T_2$ does not eliminate quantifiers.
Expand the language by unary predicates $P_n(x)$, one for each positive integer $n$, interpreted as “the $E$-class of $x$ has size $n$”. In this definitional expansion, back-and-forth on finite substructures gives quantifier elimination.
Back to article page
