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- W4214634309 abstract "We develop new techniques, for use in indestructibility arguments, of lifting embeddings on transitive set models of ZFC− which lack closure and of lifting iterations of such embeddings to a forcing extension. We use these techniques to establish basic indestructibility results for Ramsey and the Ramsey-like cardinals – α-iterable, strongly Ramsey, and super Ramsey cardinals – introduced in [11]. We show that Ramsey, α-iterable, strongly Ramsey, and super Ramsey cardinals κ are indestructible by small forcing, the canonical forcing of the GCH, and the forcing to add a fast function on κ. We also show that if κ is one of these large cardinals, then there is a forcing extension in which the large cardinal property of κ becomes indestructible by Add(κ,θ) for any cardinal θ. The following are consequences of the indestructibility results. If κ is Ramsey, α-iterable, strongly Ramsey, or super Ramsey, then there is a forcing extension preserving this in which the GCH fails at κ. If κ is one of these large cardinals, then there is a forcing extension preserving the large cardinal property of κ in which κ is not even weakly compact in HOD. If κ is Ramsey, then there is a forcing extension in which κ remains virtually Ramsey, but is no longer Ramsey (this answers positively a question posed in [11])." @default.
- W4214634309 created "2022-03-02" @default.
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- W4214634309 date "2022-06-01" @default.
- W4214634309 modified "2023-10-01" @default.
- W4214634309 title "Indestructibility properties of Ramsey and Ramsey-like cardinals" @default.
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- W4214634309 doi "https://doi.org/10.1016/j.apal.2022.103106" @default.
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