Logic Colloquium 2008
Matteo Viale Forcing axioms, supercompact cardinals, singular cardinal combinatorics
I will give a survey on my work over the combinatorics at the successor of a singular cardinal in these two different scenarios: -the singular cardinal is larger than a strongly compact cardinal $\kappa$ and of cofinality smaller than $\kappa$, -A strong forcing axiom like Martinīs maximum or the proper forcing axiom PFA holds. Under these assumptions I will show that a variety of combinatorial objects which I call covering matrices are useful tools to attack many of the classical problems on successor of singular cardinals, for example we can obtain proofs of Solovay's theorem that the singular cardinal hypothesis SCH holds above a strongly compact cardinal, or that PFA implies SCH, or a recent characterization by me and Assaf Sharon of the structure of the approachability ideal for successors of singular cardinals of the type described above, etc... Further informations available at: www.logic.univie.ac.at/~matteo