Abstract
The two classical parallel concepts of “small” sets of the real line are meagre sets and null sets. Those are equivalent to Cohen forcing and Random Real forcing for ℕℕ; in spite of this similarity, the Cohen forcing and Random Real forcing have very different shapes. One of these differences is in the fact that the Cohen forcing has an easy natural generalization for λ2 for regular λ > ℵ0, corresponding to an extension for the meagre sets, while the Random Real forcing didn’t seem to have a natural generalization, as Lebesgue measure doesn’t have a generalization for space 2λ while λ > ℵ0. The work [6] found a forcing resembling the properties of Random Real forcing for 2λ while λ is a weakly compact cardinal. Here we describe, with additional assumptions, such a forcing for 2λ while λ is just an Inaccessible Cardinal; this forcing is strategically < λ-complete and satisfies the λ+- c.c. hence preserves cardinals and cofinalities, however unlike Cohen forcing, does not add an undominated real.
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Partially sponsored by the European Research Council grant 338821. Publication 1085 on the second author’s list.
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Cohen, S., Shelah, S. Generalizing random real forcing for inaccessible cardinals. Isr. J. Math. 234, 547–580 (2019). https://doi.org/10.1007/s11856-019-1925-z
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DOI: https://doi.org/10.1007/s11856-019-1925-z