The results herein form part of a larger project to characterize the classification properties of the class of submodels of a homogeneous stable diagram in terms of the solvability (in the sense of ) of the potential isomorphism problem for this class of submodels.
We restrict ourselves to locally saturated submodels of the monster model , of some power π. We assume that in Gödel's constructive universe , π is a regular cardinal at least the successor of the first cardinal in which , is stable.
We show that the collection of pairs of submodels in as above which are potentially isomorphic with respect to certain cardinal-preserving extensions of is equiconstructible with 0#. As 0# is highly “transcendental” over , this provides a very strong statement to the effect that potential isomorphism for this class of models not only fails to be set-theoretically absolute, but is of high (indeed of the highest possible) complexity.
The proof uses a novel method that does away with the need for a linear order on the skeleton.