Suppose that a <first-order formula> $\varphi$ described the inaccessible cardinal $\kappa$, so that $\kappa$ were the least ordinal with $V_\kappa\models\varphi$. Since $\kappa$ is inaccessible, $V_\kappa$ is a <model of a first-order theory>[model] of ZFC. Apply the <Lévy reflection theorem> inside this model to the single formula $\varphi$. There is some $\alpha<\kappa$ for which
$$
V_\alpha\models\varphi
\quad\Longleftrightarrow\quad
V_\kappa\models\varphi.
$$
The right side holds, so the left side contradicts the asserted minimality of $\kappa$. Hence no first-order formula describes an inaccessible cardinal, as recorded by <ordinal described by a first-order formula>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 116 / 2 / b / Solution

A <first-order formula> $\varphi$ describes an ordinal $\alpha$ when $\alpha$ is the least ordinal such that $V_\alpha\models\varphi$. No <strongly inaccessible cardinal> can be described: if $V_\kappa\models\varphi$, the <Lévy reflection theorem> produces some $\alpha<\kappa$ with $V_\alpha\models\varphi$.

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 Incoming links: Ordinal described by a first-order formula