### Forcing Isomorphism II

by Laskowski and Shelah. [LwSh:518]

J Symbolic Logic, 1996

If T has only countably many complete types, yet has a type of
infinite multiplicity then there is a ccc forcing notion Q
such that, in any Q --generic extension of the universe, there are
non-isomorphic models M_1 and M_2 of T that can be forced
isomorphic by a ccc forcing. We give examples showing that the
hypothesis on the number of complete types is necessary and what
happens if ``ccc'' is replaced other cardinal-preserving
adjectives. We also give an example showing that membership in a
pseudo-elementary class can be altered by very simple
cardinal-preserving forcings.

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