From wikipedia:

Shoenfield's absoluteness theorem shows that $\Pi^1_2$ and $\Sigma^1_2$ sentences in the analytical hierarchy are absolute between a model $V$ of ZF and the constructible universe $L$ of the model, when interpreted as statements about the natural numbers in each model.

**Question.** Is there an "ordinal" version of the arithmetical and/or analytic hierarchies such that an analogue of Shoenfield absoluteness can be proven for that hierarchy? (With $\omega$ replaced by $\mathrm{On}$). My gut feeling is that we have to choose our language very carefully, in order to block $\aleph_1$ and/or $\beth_1$ from being definable by formulae somewhere in the hierarchy.

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