### 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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