Indecomposable ultrafilters were introduced by Keisler and Prikry as a weakening of measures on measurable cardinals, loosely speaking, by not insisting on countable completeness. Silver asked whether an inaccessible cardinal carrying an indecomposable ultrafilter necessarily has to be measurable. Sheard answered this negatively. However, recently Goldberg showed Silver's question has a positive answer above a strongly compact cardinal. We will show that strong forcing axioms, for example PFA, imply a positive answer to Silver's question. This adds to a long list of combinatorial statements giving evidence to the heuristic that strong forcing axioms assert omega_2 is ``supercompact", including failure of square principles and the singular cardinal hypothesis. Then we will define a family of weak indexed square principles and use them to demonstrate that the positive result we obtain is indeed optimal. Joint work with Chris Lambie-Hanson and Assaf Rinot.