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I know that forcing essentially does not change the ordinals, but by small I mean in comparison with other ordinals whose definition might not be stable under forcing, like the smallest uncountable ordinal $\omega_1$ or the Church-Kleene ordinal.

For example, given an ordinal in the ground model one can always make it countable in a forcing model, hence it can become smaller than $\omega_1$ after forcing (the $\omega_1$ of the forcing model of course).

My question is can we do better ? Is there some smaller class (downward closed) of ordinals such that any ordinal can be "put into this class" by forcing ?

Typically: can every ordinal become a recursive ordinal after forcing ? (i.e. smaller than the Church-Kleene ordinal)

Or is there obstructions preventing such a thing for any class of ordinals smaller than the class of all countable ordinals ?

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    $\begingroup$ You definitely cannot make a nonrecursive ordinal recursive by forcing. Being recursive is witnessed by a Turing machine, i.e. a natural number, and these aren't affected by forcing. $\endgroup$
    – Miha Habič
    Commented Sep 2, 2015 at 12:18
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    $\begingroup$ You can change the set of ordinals which are countable in HOD by forcing, and this set can be smaller than $\omega_1^V$. $\endgroup$
    – Yair Hayut
    Commented Sep 3, 2015 at 6:54

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