What are mathematical theories? What mathematical objects should
correspond to this informal concept? The classical and most
important answer to these questions is: Theories formalized in
first order logic. But this answer has also some undesirable
features. One of theme is the dependence of such theories upon the
language or the choice of primitive concepts, whereas a slightly
deeper view would identify theories interpretable in each other.
The purpose of the present memoir is to investigate further, to
survey the former work and to point out a number of open problems
about local interpretability.
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