Suppose I enlarge the first-order logic with an "almost all" quantifier, let's denote it by G, ie.:
$G_x P(x) \iff$ for all but finitely many x, P(x)
Syntax for G is the same as for other quantifiers.
Suppose I am working over the first-order theory of natural numbers. For every sentence $T$ using $G$, does there exist a sentence $S$ over "standard" first-order logic, such that $S \iff T$ in every countable model of natural numbers?