Timeline for Failure of the GCH
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Dec 15, 2017 at 3:21 | comment | added | Keith Millar | Note that a $P^2(\kappa)$-hypermeasurable is the same is a $\kappa+2$-strong. | |
| Nov 4, 2011 at 20:19 | comment | added | Monroe Eskew | SCH must hold above a strongly compact cardinal, so there can be no strong compactness in a model of global failure of GCH. | |
| Nov 3, 2011 at 22:06 | comment | added | eiths | Thanks a lot for the answers and the references. So, by Andres' comment, we know the exact large cardinal assumption (hypermeasurability) leading to a full control of the degree of failure of GCH. Now, do we also know what kind of large cardinals can survive in these models where GCH fails? Moreover, what if we are generous enough to assume more than what is actually needed for global failure of GCH e.g., say a supercompact. Can we then get larger cardinals together with GCH failure? | |
| Nov 3, 2011 at 21:55 | vote | accept | eiths | ||
| Nov 3, 2011 at 19:12 | comment | added | Andrés E. Caicedo | Timothy, The unpublished proofs are by now standard. You can get $ZFC+\forall\kappa(2^\kappa=\kappa^{+n})$ for any fixed natural $n>0$ from ${\mathcal P}^n(\kappa)$-hypermeasurables. The arguments in, for example, J. Cummings paper (on violating SCH at all limits) or the many papers by Gitik and his students (including his very nice Handbook article) should explain how to fill in any missing details. | |
| Nov 3, 2011 at 18:03 | history | answered | Timothy Chow | CC BY-SA 3.0 |