1998/04/15 by Shelah, Saharon
#FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.math/9804156
The original theme of the paper is the existence proof of ``there is < etaalpha : alpha < lambda > which is a (lambda,J)-sequence for < Ii:i, a sequence of ideals. This can be thought of as in a generalization to Luzin sets and Sierpinski sets, but for the product prodi< delta Dom(Ii), the existence proofs are related to pcf. The second theme is when does a Boolean algebra B has free caliber lambda (i.e. if X subseteq B and |X|= lambda, then for some Y subseteq X with |Y|= lambda and Y is independent). We consider it for B being a Maharam measure algebra, or B a (small) product of free Boolean algebras, and kappa-cc Boolean algebras. A central case lambda = (bethomega)+ or more generally, lambda = mu+ for mu strong limit singular of ``small'' cofinality. A second one is mu = mu< kappa< lambda < 2mu ; the main case is lambda regular but we also have things to say on the singular case. Lastly, we deal with ultraproducts of Boolean algebras in relation to irr(-) and s(-) etc.