  Model-theoretic aspects of exotic smoothness were studied long ago uncovering
unexpected relations to noncommutative spaces and quantum theory. Some of these
relations were worked out in detail in later work. An important point in the
argumentation was the forcing construction of Cohen but without a direct
application to exotic smoothness. In this article we assign the set-theoretic
forcing on trees to Casson handles and characterize small exotic smooth $R^4$
from this point of view. Moreover, we show how models in some Grothendieck
toposes can help describing such differential structures in dimension 4. These
results can be used to obtain the deformation of the algebra of usual complex
functions to the noncommutative algebra of operators on a Hilbert space. We
also discuss the results in the context of the Epstein-Glaser renormalization
in QFT.
