A parallel to the null ideal for inaccessible $\lambda$: Part I

by Shelah. [Sh:1004]
Archive for Math Logic, 2017
It is well known how 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 naturally, to call it a generalization we require it to be (< lambda)-complete and lambda^+-c.c. and more. Of course, we would welcome additional properties generalizing the ones of the random real forcing. Returning to the ideal (instead of forcing) we may look at the Boolean Algebra of lambda-Borel sets modulo the ideal. Common wisdom have said that there is no such thing, but here surprisingly underline {first} we get a positive = existence answer for lambda a ``mild'' large cardinal: the weakly compact one. Second, we try to show that this together with the meagre ideal (for lambda) behaves as in the countable case. In particular, consider the classical Cichon diagram, which compare several cardinal characterizations of those ideals. Last but not least, we apply this to get consistency results on cardinal invariants for such lambda 's. We shall deal with other cardinals, and with more properties related forcing notions in a continuation.

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