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

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 because we have no parallel of Lebesgue integral, but here surprisingly first we get a positive $$=$$ existence answer for a generalization of the null ideal for a “mild” large cardinal $$\lambda $$ —a 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, we consider the classical Cichoń diagram, which compares several cardinal characterizations of those ideals. We shall deal with other cardinals, and with more properties of related forcing notions in subsequent papers (Shelah in The null ideal for uncountable cardinals; Iterations adding no $$\lambda $$ -Cohen; Random $$\lambda $$ -reals for inaccessible continued; Creature iteration for inaccesibles. Preprint; Bounding forcing with chain conditions for uncountable cardinals) and Cohen and Shelah (On a parallel of random real forcing for inaccessible cardinals. arXiv:1603.08362 [math.LO]) and a joint work with Baumhauer and Goldstern.