= Solution
By the standard interpretation of the representing <logical formula>,
$$
\mathbb N\models G_S\quad\Longleftrightarrow\quad S\nvdash G_S.
$$
If $S$ is <first-order consistent>, part (b) supplies the right-hand side, so $G_S$ is true. If $S$ is inconsistent, explosion gives $S\vdash G_S$, so the representing existential statement is true and $G_S$ is false. Therefore
$$
\boxed{\mathbb N\models G_S\quad\Longleftrightarrow\quad S\text{ is consistent}.}
$$
This is an external truth assertion about the standard <natural numbers>. Formalizing that assertion inside a <first-order theory> requires the additional facts explicitly supplied in part (d).
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