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Now it seems this definition would not be invariant under forcing. If we ask for a definition of truth in $L$ which is invariant under forcing, would that require $0^{\#}$ to exist?
Are you computing the complexity of $t \subset \omega$ or of $\{t\} \subset \mathcal{R}?$ In the case of the former, $\Pi_3^1$ would follow from $\Sigma_3^1$ simply by checking if $L \models \neg \varphi.$
You're right, the reason he made the definition this way was to create a predicate for truth with parameters in $L.$ So I guess the question is, for truth without parameters, can we make it $\Delta_n^1$ for some $n?$
