# Sh:1004

- Shelah, S. (2017).
*A parallel to the null ideal for inaccessible \lambda: Part I*. Arch. Math. Logic,**56**(3-4), 319–383. arXiv: 1202.5799 DOI: 10.1007/s00153-017-0524-0 MR: 3633799 -
Abstract:

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 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 Cichoń 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. - published version (65p)

Bib entry

@article{Sh:1004, author = {Shelah, Saharon}, title = {{A parallel to the null ideal for inaccessible $\lambda$: Part I}}, journal = {Arch. Math. Logic}, fjournal = {Archive for Mathematical Logic}, volume = {56}, number = {3-4}, year = {2017}, pages = {319--383}, issn = {0933-5846}, doi = {10.1007/s00153-017-0524-0}, mrclass = {03E35 (03E55)}, mrnumber = {3633799}, mrreviewer = {Chris Lambie-Hanson}, doi = {10.1007/s00153-017-0524-0}, note = {\href{https://arxiv.org/abs/1202.5799}{arXiv: 1202.5799}}, arxiv_number = {1202.5799} }