We develop several aspects of local and global stability in continuous first order logic. In particular, we study type-definable groups and genericity.

Let G be a group definable in a monster model of a rosy theory satisfying NIP. Assume that G has hereditarily finitely satisfiable generics and 1 < U b(G) < ∞. We prove that if G acts definably on a definable set of U р-rank 1, then, under some general assumption about this action, there is an infinite field interpretable in . We conclude that if G is not solvable-by-finite and it acts faithfully and definably on a definable set of U р-rank 1, then there is an infinite field interpretable in . As an immediate consequence, we get that if G has a definable subgroup H such that U р(G) = U р(H) + 1 and G/⋂g∈G Hg is not solvable-by-finite, then an infinite field interpretable in also exists.

We prove a number of results in effective randomness, using methods in which Π1 0 classes play an essential role. The results proved include the fact that every PA Turing degree is the join of two random Turing degrees, and the existence of a minimal pair of LR degrees below the LR degree of the halting problem.