In this paper we introduce a tree-like forcing notion extending some properties of the random forcing in the context of $$2^\kappa $$ 2 κ , $$\kappa $$ κ inaccessible, and study its associated ideal of null sets and notion of measurability. This issue was addressed by Shelah (On CON(Dominating $$\_$$ _ lambda $$\,>\,$$ > cov $$\_\lambda $$ _ λ (meagre)), arXiv:0904.0817 , Problem 0.5) and concerns the definition of a forcing which is $$\kappa ^\kappa $$ κ κ -bounding, $$<\kappa $$ < κ -closed and $$\kappa ^+$$ κ + -cc, for $$\kappa $$ κ inaccessible. Cohen and Shelah (Generalizing random real forcing for inaccessible cardinals, arXiv:1603.08362 ) provide a proof for (Shelah, On CON(Dominating $$\_$$ _ lambda $$\,>\,$$ > cov $$\_\lambda $$ _ λ (meagre)), arXiv:0904.0817 , Problem 0.5), and in this paper we independently reprove this result by using a different type of construction. This also contributes to a line of research adressed in the survey paper (Khomskii et al. in Math L Q 62(4–5):439–456, 2016).
Archive for Mathematical Logic – Springer Journals
Published: Jun 19, 2017
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