## 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

### MSC:

 3e+35 Consistency and independence results 3e+55 Large cardinals

### Keywords:

supercompact cardinal; proper forcing

Zbl 0437.03026
Full Text:

### References:

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