Generalizing random real forcing for inaccessible cardinals

Shani Cohen*, Saharon Shelah

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

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 ℕ; 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.

Original languageEnglish
Pages (from-to)547-580
Number of pages34
JournalIsrael Journal of Mathematics
Volume234
Issue number2
DOIs
StatePublished - 1 Oct 2019

Bibliographical note

Publisher Copyright:
© 2019, The Hebrew University of Jerusalem.

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