In this article the structure of the models of decidable monadic theories of planar graphs is investigated. It is shown that if the monadic theory of a class K of planar graphs is decidable, then the tree-width in the sense of Robertson and Seymour of the elements of K is universally bounded and there is a class T of trees such that the monadic theory of K is interpretable in the monadic theory of T.

Assume ZFC is consistent then for everyB⫅ω there is a generic extension of the ground world whereB is recursive in the monadic theory ofω 2.