A null ideal for inaccessibles
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In this paper we introduce a tree-like forcing notion extending some properties of the random forcing in the context of \(2^\kappa \), \(\kappa \) inaccessible, and study its associated ideal of null sets and notion of measurability. This issue was addressed by Shelah (On CON(Dominating\(\_\)lambda\(\,>\,\)cov\(\_\lambda \)(meagre)), arXiv:0904.0817, Problem 0.5) and concerns the definition of a forcing which is \(\kappa ^\kappa \)-bounding, \(<\kappa \)-closed and \(\kappa ^+\)-cc, for \(\kappa \) inaccessible. Cohen and Shelah (Generalizing random real forcing for inaccessible cardinals, arXiv:1603.08362) provide a proof for (Shelah, On CON(Dominating\(\_\)lambda\(\,>\,\)cov\(\_\lambda \)(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).
KeywordsGeneralized Baire space Generalized random forcing Null ideal
Mathematics Subject Classification03-XX (Mathematical logic and foundations)
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