Source: SHAWN HEDMAN
Definition: Let $M$ and $N$ be $V$-structures. If $M$ and $N$ models the same $V$-sentences, then $M$ and $N$ are said to be elementarily equivalent, denoted $M \equiv N.$
Example: the $V_{<}$-structures $\mathbb{Q}_{<}$ and $\mathbb{R}_{<}$ are elementarily equivalent.
My question: Is not this sentence: $\forall x((x = \pi) \implies$ (x =x))? and the second structure models it? While the first one doesn't as $\pi$ not in $\mathbb{Q}$!
Is $\pi$ even a symbol in $V$? If not this is not a sentence to begin with.
Note that if $\pi$ is in the language then it has to have some interpretation in $\Bbb Q$, but that won't be the same number you think about, but rather some rational number.