Potential isomorphism of elementary substructures of a strictly stable homogeneous model

Abstract

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 [1]) of the potential isomorphism problem for this class of submodels.

We restrict ourselves to locally saturated submodels of the monster model $\mathfrak{M}$ of some power π. We assume that in Gödel's constructible universe 𝕃, π is a regular cardinal at least the successor of the first cardinal in which $\mathfrak{M}$ is stable.

We show that the collection of pairs of submodels in 𝕃 as above which are potentially isomorphic with respect to certain cardinal-preserving extensions of 𝕃 is equiconstructible with 0^#. As 0^# is highly “transcendental” over 𝕃, 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.