1- What are examples of statements that are indeterminate from $\sf ZFC + [V=L]$ that are comparable to the situation of $\sf CH$ with regards to $\sf ZFC$. I mean I need a statement $\psi$ such that $\sf ZFC + [V=L]$ is equi-consistent with the addition of $\psi$ or $\neg \psi$ to it. That is, a statement that both itself an its negation doesn't increase the consistency strength of $\sf ZFC +[V=L]$ when added to it. Are there some simple natural statements. In particular, are there some statements regarding cardinal comparisons?
I'll add another related question, that is:
2- if we instead of the simple natural requirement, replace it by $\psi$ meeting the following two conditions:
$\sf (ZFC+[V=L] + \psi) \not \vdash Con(ZFC) \\ \sf (ZFC+[V=L]+ \psi) \not \vdash \neg Con(ZFC)$