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

Suppose that a first-order formula $\phi$ described the inaccessible cardinal $\kappa$, so that $\kappa$ were the least ordinal with $V_{\kappa }\models \phi$. Since $\kappa$ is inaccessible, $V_{\kappa }$ is a model of ZFC. Apply the Lévy reflection theorem inside this model to the single formula $\phi$. There is some $\alpha <\kappa$ for whichThe 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.