SAT solvers for use in $P^{NP}$ and $NP^{NP}$ The original question I am answering:

Can you use SAT solvers to solve problems complete in $\Sigma_2^P,\Pi_2^P,\Delta_2^P$?

My first thought:

Venn Diagrams show that the PH encloses NP, but does not equal NP. Therefore, there should be some problems that cannot be solvable using SAT?

My second thought:

To be able to use a SAT solver on one of those PH classes, it would imply that they are reducible to SAT, and so $\Sigma_2^P = \Pi_2^P = \Delta_2^P = \mathsf{NP}$.

This would imply that PH collapses to NP, which we know not to be true.
Are these thoughts correct?
We don't know that PH doesn't collapse to NP. We don't even know that PH doesn't collapse all the way to P.
The best you can say is that you can use SAT solvers to solve problems in one of these classes iff PH collapses to NP.