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 < Ub(G) < ∞. We prove that if G acts definably on a definable set of Uр-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 Uр-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 Uр(G) = Uр(H) + 1 and G/⋂g∈GHg 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 Π10 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.