### Potential isomorphism and semi--proper trees

by Hellsten and Hyttinen and Shelah. [HHSh:770]

Fundamenta Math, 2002

We study a notion of potential isomorphism, where two
structures are said to be potentially isomorphic if they are
isomorphic in some generic extension that preserves stationary sets
and does not add new sets of cardinality less than the cardinality
of the models. We introduce the notions of semi-proper and weakly
semi-proper trees, and note that there is a strong connection
between the existence of potentially isomorphic models for a given
complete theory and the existence of weakly semi-proper trees.
We prove the existence of semi-proper trees under certain cardinal
arithmetic assumptions. We also show the consistency of the
non-existence of weakly semi-proper trees assuming the consistency
of some large cardinals.

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