Abstract
It is known that if \(\kappa < \lambda\) are such that κ is indestructibly supercompact and λ is 2λ supercompact, then level by level equivalence between strong compactness and supercompactness fails. We prove a theorem which points towards this result being best possible. Specifically, we show that relative to the existence of a supercompact cardinal, there is a model for level by level equivalence between strong compactness and supercompactness containing a supercompact cardinal κ in which κ’s strong compactness is indestructible under κ-directed closed forcing.
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The author’s research was partially supported by PSC-CUNY Grant 66489-00-35 and a CUNY Collaborative Incentive Grant.
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Apter, A.W. Supercompactness and level by level equivalence are compatible with indestructibility for strong compactness. Arch. Math. Logic 46, 155–163 (2007). https://doi.org/10.1007/s00153-007-0034-6
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DOI: https://doi.org/10.1007/s00153-007-0034-6
Keywords
- Supercompact cardinal
- Strongly compact cardinal
- Indestructibility
- Gitik iteration
- Prikry forcing
- Non-reflecting stationary set of ordinals
- Level by level equivalence between strong compactness and supercompactness