The dichotomy conjecture for Polish structures

The paper considers definable structures on the reals, particularly Polish spaces and algebras, and compares their models across forcing extensions. A model is called categorical when corresponding interpretations in forcing extensions with the same continuum are isomorphic; a model is called free when its interpretations are free algebras.

Dichotomy conjecture. There is a dichotomy: either the model is similar to categorical theories, or there are “many complicated models”.

This is proposed as part of a possible stability or classification theory for Polish spaces and more general definable structures. The supplied text gives no resolution of the dichotomy.