Precipitousness in forcing extensions. (English) Zbl 0595.03044

W. Mitchell [see T. Jech, M. Magidor, W. Mitchell, and K. Prikry, J. Symb. Logic 45, 1-8 (1980; Zbl 0437.03026)] proved that if a measurable cardinal \(\kappa\) is collapsed to \(\omega_ 1\) by the Lévy algebra \({\mathcal L}^{\kappa}\), then \(\kappa =\omega_ 1\) is precipitous in the extension. The author introduces a large class \({\mathcal A}\) of proper forcings containing, e. g., the standard poset for making the proper forcing axiom true. The main result says that assuming \(\kappa\) supercompact, \({\mathcal L}^{\kappa}\Vdash P\in {\mathcal A}\), then \({\mathcal L}^{\kappa}*P\Vdash\) \(''\omega_ 1\) is precipitous”. As a corrollary, Con(ZFC \(+\) two supercompact cardinals) implies Con(ZFC \(+\) proper forcing axiom \(+\) \(\omega_ 1\) precipitous).
Reviewer: L.Bukovský


03E35 Consistency and independence results
03E55 Large cardinals


Zbl 0437.03026
Full Text: DOI


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