 True Arithmetic (\(\textrm{TA}\)) corresponds to the set of Gödel numbers of first-order arithmetical sentences true in the standard model of \(\mathcal{L}_a\)—i.e., \(\textrm{TA} = \{\ulcorner \varphi \urcorner : \varphi \in \mathcal{L}_a \ \wedge \ \mathfrak{N} \models \varphi\}\). Prior to the definition of the analytical hierarchy itself, Hilbert &amp; Bernays had already showed the following: