Many forcing axioms for all regular uncountable cardinals

Abstract A central theme in set theory is to find universes with extreme, well-understood behaviour. The case we are interested in is assuming GCH and having a strong forcing axiom of higher order than usual. Instead of “every suitable forcing notion of size λ has a sufficiently generic filter” we shall say “for every suitable method of producing notions of forcing based on a given stationary set, there is such a suitable stationary set S and sufficiently generic filters for the notion of forcing attached to S ”. Such notions of forcing are important for Abelian group theory, but this application is delayed for a sequel.