First I will define and explain the "Borel conjecture" and the "Dual Borel conjecture", then I will present two forcing notions: Ultralaver forcing, a variant of Laver forcing which uses ultrafilters Janus forcing, which—depending on your point of view—can be seen as a variant of Cohen forcing or of random forcing. Ultralaver forcing can be used to show the consistency of the Borel conjecture (Laver's original proof used Laver forcing). Janus forcing can be used to show the consistency of the dual Borel conjecture (Carlson's original proof used Cohen forcing). The main point of these two new forcing notions is that they can be used to show the joint consistency of Borel plus dual Borel conjecture. (This is done using a strange kind of iteration; the iteration will be presented in a later talk next semester.)

This is joint work with J. Kellner, S. Shelah and W. Wohofsky.