We complete the proof of Shelah's theorem on properness preservation under countable support iterations.

We finish the discussion of two-step iteration of proper forcing.

The primary goal of this talk is to introduce two equivalent definitions of algorithmically random sequences, one given in terms of a specific collection of effective statistical tests (known as Martin-Löf tests) and another given in terms of initial segment complexity (i.e., Kolmogorov complexity). I will explain how these definitions can be generalized to hold for various computable probability measures on Cantor space, and if time permits, I will discuss recent work with Rupert Hölzl and Wolfgang Merkle in which we study the interplay between (i) the growth rate of the initial segment complexity of sequences random with respect to some computable probability measure and (ii) certain properties of this underlying measure (such as continuity vs. discontinuity).  No background in algorithmic randomness will be assumed.

Starting from a 2-huge cardinal, we construct a model where for all pairs of regular cardinals kappa<lambda, (lambda^+,lambda) --> (kappa^+,kappa) and there is a lambda^+ saturated ideal on P_{kappa^+}(lambda).  Then using a modified Radin forcing we get similar global principles involving singular cardinals but with only finite jumps.

We finish the discussion of the propagation of genericity of conditions in two-step iterations, in preparation for the proof of the proper forcing iteration theorem.