Let $S:=\{ \alpha\in \aleph_1 \mid \exists M, M \models ZFC \wedge M\cap \textbf{ORD} =\alpha \wedge |M|=\aleph_0 \}$. Has there been any research into the properties of S?
In particular, I'm interested in the connection to Shoenfield Absoluteness. Sense any two models of set theory containing the same ordinals agree on all $\Sigma_2^1$ and $\Pi_2^1$ arithmetical sentences, forcing is insufficient to prove independence of such statements. It seems to me that the logical first place to start is exploring this S. I've found a proof that this S is unbounded, but anything more seems incredibly difficult.