Sacks forcing allows us to build a model $V[G]$, such that there is no "intermediate model" between $V$ and $V[G]$, meaning if $V \subseteq W \subseteq V[G]$ is a model of ZFC then either $W = V$ or $W = V[G]$.
My question is:
- Whether we know how to force a model to have exactly $1$ intermediate model?
- Assuming the answer is Yes, do we know how to extend this result arbitrarily long? Meaning, do we know how to force a model where all the inner models (i.e standard, transitive classes that contain all the ordinals) are well ordered and have a $1-1$ correspondence with the ordinals?
This would necessate to start with $V_0 = L$, and then possibly take $V_1 = V_0[G]$ (where this would be a regular Sacks forcing) and somehow continue to create $V_\alpha$ for all $\alpha \in ORD$, without 'accidentally' creating any more inner models.