Forcing Isomorphism

If two models of a first order theory are isomorphic then they remain isomorphic in any forcing extension of the universe of sets. In general, however, such a forcing extension may create new isomorphisms. For example, any forcing that collapses cardinals may easily make formerly non-isomorphic models isomorphic. Certain model theoretic constraints on the theory and other constraints on the forcing can prevent this pathology. A countable first order theory is said to be classifiable if it is superstable and does not have either the dimensional order property (DOP) or the omitting types order property (OTOP). Shelah has shown [7] that if a theory T is classifiable then each model of cardinality λ is described by a sentence of