A parallel to the null ideal for inaccessible lambda. Part I

Saharon Shelah

Published 2012-02-26, updated 2017-01-18Version 3

It is well known to generalize the meagre ideal replacing aleph_0 by a (regular) cardinal lambda > aleph_0 and requiring the ideal to be lambda^+-complete. But can we generalize the null ideal? In terms of forcing, this means finding a forcing notion similar to the random real forcing, replacing aleph_0 by lambda, so requiring it to be (<lambda)-complete. Of course, we would welcome additional properties generalizing the ones of the random real forcing. Returning to the ideal (instead forcing) we may look at the Boolean Algebra of lambda-Borel sets modulo the ideal. Surprisingly we get an positive = existence answer for lambda a "mild" large cardinals: the weakly compact ones. We apply this to get consistency results on cardinal invariants for such lambda's. We shall deal with other cardinals more properties related forcing notions in a continuation.