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.



Is there an analogue of Shoenfield's absoluteness theorem, but for $\mathrm{On}$?
Perhaps the solution lies in the Levy absoluteness theorem.


We define the Levy hierarchy of formulas as follows.


1. $\Sigma\_0,\Pi\_0,\Delta\_0$ are formulas in the language of set theory which only have bounded quantifiers.
2. $\Sigma\_{n+1}$ formulas equivalent to formulas of the form $\exists x\varphi$, where $\varphi$ is $\Pi\_n$; and similarly $\Pi\_{n+1}$ are formulas equivalent to $\forall x\varphi$ where $\varphi$ is $\Sigma\_n$.
3. $\Delta\_n$ are formulas which are both $\Sigma\_n$ and $\Pi\_n$.


Now we have that if $M$ is a transitive class (set or otherwise), then $\Delta\_0$ formulas are absolute between $M$ and $V$ (as long as $M$ includes all the parameters). Next we have that $\Sigma\_1$ formulas are upwards absolute and $\Pi\_1$ formulas are downwards absolute, so $\Delta\_1$ formulas are absolute.


