Abstract
Say that \({\kappa}\)’s measurability is destructible if there exists a < \({\kappa}\)-closed forcing adding a new subset of \({\kappa}\) which destroys \({\kappa}\)’s measurability. For any δ, let λ δ =df The least beth fixed point above δ. Suppose that \({\kappa}\) is indestructibly supercompact and there is a measurable cardinal λ > \({\kappa}\). It then follows that \({A_{1} = \{\delta < \kappa \mid \delta}\) is measurable, δ is not a limit of measurable cardinals, δ is not δ + strongly compact, and δ’s measurability is destructible when forcing with partial orderings having rank below λ δ } is unbounded in \({\kappa}\). On the other hand, under the same hypotheses, \({A_{2} = \{\delta < \kappa \mid \delta}\) is measurable, δ is not a limit of measurable cardinals, δ is not δ + strongly compact, and δ′s measurability is indestructible when forcing with either Add(δ, 1) or Add(δ, δ +)} is unbounded in \({\kappa}\) as well. The large cardinal hypothesis on λ is necessary, as we further demonstrate by constructing via forcing two distinct models in which either \({A_{1} = \emptyset}\) or \({A_{2} = \emptyset}\). In each of these models, both of which have restricted large cardinal structures above \({\kappa}\), every measurable cardinal δ which is not a limit of measurable cardinals is δ + strongly compact, and there is an indestructibly supercompact cardinal \({\kappa}\). In the model in which \({A_{1} = \emptyset}\), every measurable cardinal δ which is not a limit of measurable cardinals is <λ δ strongly compact and has its <λ δ strong compactness (and hence also its measurability) indestructible when forcing with δ-directed closed partial orderings having rank below λ δ . The choice of the least beth fixed point above δ is arbitrary, and other values of λ δ are also possible.
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This paper is dedicated to the memory of Rich Laver, a friend and inspiration. It is truly a privilege to be able to contribute a paper to this volume in his honor.
The author’s research was partially supported by PSC-CUNY grants.
The author wishes to thank the referee for helpful comments and suggestions which have been incorporated into the current version of the paper.
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Apter, A.W. Indestructibility and destructible measurable cardinals. Arch. Math. Logic 55, 3–18 (2016). https://doi.org/10.1007/s00153-015-0470-7
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DOI: https://doi.org/10.1007/s00153-015-0470-7
Keywords
- Supercompact cardinal
- Strongly compact cardinal
- Indestructibility
- Superdestructibility
- Lottery sum
- Nonreflecting stationary set of ordinals