### Forcing Isomorphism

by Baldwin and Laskowski and Shelah. [BLSh:464]

J Symbolic Logic, 1993

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.

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