Generalizing random real forcing for inaccessible cardinals

by Cohen and Shelah. [CnSh:1085]

The two 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 aleph^{aleph_0}_0 ; 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 lambda^{lambda} while lambda > aleph_0, corresponding to an extension for the meagre sets, while the Random real forcing didn't see to have a natural generalization, as Lebesgue measure doesn't have a generalization for space lambda^lambda while lambda > aleph_0 . In work [1], Shelah found a forcing resembling the properties of Random Real Forcing for lambda^lambda while lambda is a weakly compact cardinal. Here we describe, with additional assumptions, such a forcing for lambda^lambda while lambda is an inaccessible cardinal; this forcing preserves cardinals and cofinalities, however unlike Cohen forcing, does not add dominating reals.

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