User blog comment:DontDrinkH20/Some ZFC-Definable very big LCOs - uncomputable?/@comment-35470197-20180830044905/@comment-35470197-20180830224612

Oh, really? Then I am sorry. Could you tell me a proof?

Even if you quatify all FOST formulae, the rage of formulae which appears in the definition of \(\theta_{\alpha}\) heavily depends on the choice of formalisation, and hence it is not obvious for me.

The former one considers all witness at \(V_{\alpha}\) and all ordinals in \(V_{\alpha}\), while the latter one ocnsiders all witness at \(V_{\alpha} \cap \beta\) and all odinals \(\beta\).

The situation will get more complicated if you consider transitive models \(M\), which is not necessarily an \(\omega\)-model.