Destruction or preservation as you like it

Annals of Pure and Applied Logic 91 (2-3):191-229 (1998)

Authors
Joel David Hamkins
Oxford University
Abstract
The Gap Forcing Theorem, a key contribution of this paper, implies essentially that after any reverse Easton iteration of closed forcing, such as the Laver preparation, every supercompactness measure on a supercompact cardinal extends a measure from the ground model. Thus, such forcing can create no new supercompact cardinals, and, if the GCH holds, neither can it increase the degree of supercompactness of any cardinal; in particular, it can create no new measurable cardinals. In a crescendo of what I call exact preservation theorems, I use this new technology to perform a kind of partial Laver preparation, and thereby finely control the class of posets which preserve a supercompact cardinal. Eventually, I prove the ‘As You Like It’ Theorem, which asserts that the class of κ -directed closed posets which preserve a supercompact cardinal κ can be made by forcing to conform with any pre-selected local definition which respects the equivalence of forcing. Along the way I separate completely the levels of the superdestructibility hierarchy, and, in an epilogue, prove that the notions of fragility and superdestructibility are orthogonal — all four combinations are possible
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DOI 10.1016/s0168-0072(97)00044-4
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References found in this work BETA

Fragile Measurability.Joel Hamkins - 1994 - Journal of Symbolic Logic 59 (1):262-282.
Canonical Seeds and Prikry Trees.Joel David Hamkins - 1997 - Journal of Symbolic Logic 62 (2):373-396.
Laver Indestructibility and the Class of Compact Cardinals.Arthur W. Apter - 1998 - Journal of Symbolic Logic 63 (1):149-157.
Small Forcing Makes Any Cardinal Superdestructible.Joel David Hamkins - 1998 - Journal of Symbolic Logic 63 (1):51-58.

View all 6 references / Add more references

Citations of this work BETA

The Lottery Preparation.Joel David Hamkins - 2000 - Annals of Pure and Applied Logic 101 (2-3):103-146.
Gap Forcing: Generalizing the Lévy-Solovay Theorem.Joel David Hamkins - 1999 - Bulletin of Symbolic Logic 5 (2):264-272.
Tall Cardinals.Joel D. Hamkins - 2009 - Mathematical Logic Quarterly 55 (1):68-86.
The Large Cardinals Between Supercompact and Almost-Huge.Norman Lewis Perlmutter - 2015 - Archive for Mathematical Logic 54 (3-4):257-289.

View all 11 citations / Add more citations

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