This concerns the (clarified) notion of being strongly unreachable from the outline on HP you circulated yesterday. (M is strongly unreachable if for all proper inner models N of M, for all sufficiently large M-cardinals as computed in N is strictly less than as computed in M).
Suppose V is strongly unreachable (and just relative to -definable classes from parameters to make this explicitly first order). Then there are no measurable cardinals and either
- V = HOD and in fact V = K, (so GCH holds, and much more); or
- global-PD holds.
(K refers to a natural generalization of the usual core model—the union of “lower-parts of structures”—and this could be L of course. This K must be very L-like because of having no measurable cardinals. Global-PD is the assertion that PD holds in all set-generic extensions).
These are not mutually exclusive possibilities. But I actually do not know if (2) is possible. This leads to some rather subtle questions about correctness, for example suppose that M is countable and M is the minimum correct model of ZFC+global-PD. Must M be strongly unreachable? It seems likely that the answer should be yes, but this looks quite difficult (to me anyway).
(“Correct” here means that the set-generic extensions of M are projectively correct)