Matteo Viale — Strong forcing axioms and the continuum problem , following Aspéro' and Schindler's proof that MM++ implies Woodin's Axiom (*)

A topological approach to forcing axioms considers them as strong forms of the Baire category theorem; an algebraic approach describes certain properties of "algebraic closure" for the universe of sets that can be derived from them. Our goal is to show how the theorem of Aspéro and Schindler links the geometric and algebraic points of view. Drawing on Gödel’s program, we connect these mathematical results to the philosophical debate on what could constitute a viable solution of the continuum problem.