Partial Morleyizations

Absolute model companionship, forcibility, and the continuum problem

We use absolute model companionship (amc) to study the continuum problem and to gauge the expressive power of forcing.We show that (a definable version of) is the unique solution to the continuum problem which can be in the amc of a"partial morleyization of the are class many supercompact cardinals".We also show that (assuming large cardinals)forcibility overlaps with the apparently weaker notion of consistency for any problem expressible as a of a (very large fragment of) third order arithmetic the suslin hypothesis, the whitehead conjecture for free groups and the whitehead conjecture for free groups are a small sample of such problems partial morleyizations can be described as follows : let be the set of first order for is the expansion of adding atomic relation symbols for all formulae in and is the asserting that each is logically equivalent to the corresponding atomicformula for a is the partial morleyization of induced by is the strengthening of model companionship defined as follows : for a theory denotes the logical consequences of which are boolean combinations of universalsentences.