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A lifting argument for the generalized Grigorieff forcing. (English) Zbl 1350.03036

In this paper, the authors study a generalized version of Grigorieff forcing at inaccessible cardinals and use it to present a new proof of Woodin’s celebrated result for forcing the failure of GCH at a measurable cardinal from optimal hypotheses.
The forcing, like the case of S.-D. Friedman and K. Thompson [J. Symb. Log. 73, No. 3, 906–918 (2008; Zbl 1160.03035)], is more uniform than Woodin’s original proof (which was based on Cohen forcing), in the sense that the required guiding generic is obtained directly without going to some further extension of the universe; but it is different from that of Friedman-Thompson, as it does not have a treelike structure. Also, unlike Sacks forcing at an inaccessible which is minimal, the resulting generalized Grigorieff forcing is not minimal.
The authors think that the method might be useful for obtaining new results concerning cardinal invariants at uncountable regular cardinals (see the open questions at the end of the paper).

MSC:

03E35 Consistency and independence results
03E55 Large cardinals

Citations:

Zbl 1160.03035
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References:

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