  A forcing extension may create new isomorphisms between two models of a first
order theory. Certain model theoretic constraints on the theory and other
constraints on the forcing can prevent this pathology. A countable first order
theory is classifiable if it is superstable and does not have either the
dimensional order property or the omitting types order property. Shelah [Sh:c]
showed that if a theory T is classifiable then each model of cardinality lambda
is described by a sentence of L_{infty, lambda}. In fact this sentence can be
chosen in the L^*_{lambda}. (L^*_{lambda} is the result of enriching the
language L_{infty, beth^+} by adding for each mu < lambda a quantifier saying
the dimension of a dependence structure is greater than mu .) The truth of such
sentences will be preserved by any forcing that does not collapse cardinals <=
lambda and that adds no new countable subsets of lambda. Hence, if two models
of a classifiable theory of power lambda are non-isomorphic, they are
non-isomorphic after a lambda-complete forcing. Here we show that the
hypothesis of the forcing adding no new countable subsets of lambda cannot be
eliminated. In particular, we show that non-isomorphism of models of a
classifiable theory need not be preserved by ccc forcings.
