Timeline for Which $L$-like principles are known to be relatively consistent with large cardinals?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Jun 19 at 1:37 | history | bounty ended | Elliot Glazer | ||
| Jun 12 at 2:02 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Jun 12 at 1:47 | comment | added | Joel David Hamkins | Wow, thanks! You can get GCH, V=HOD, and ground axiom simultaneously, by first forcing GCH, and then force V=HOD using the $\Diamond^*_\kappa$ coding, and then observe that this also forces the ground axiom, using the methods of my paper with Woodin and Reitz. | |
| Jun 12 at 1:42 | comment | added | Elliot Glazer | Bounty awarded! Are there any obstructions to trying to get all the listed L-like principles at once? | |
| Jun 12 at 1:38 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Jun 12 at 1:35 | vote | accept | Elliot Glazer | ||
| Jun 12 at 1:34 | comment | added | Joel David Hamkins | Yes, because you can use $\Diamond^*_\kappa$ coding instead of GCH coding to get V=HOD. This method is detailed in work of Andrew Brooke-Taylor. Indeed, preserving the GCH is part of the point of this alternative coding method. (But this isn't indestructibility, but just V=HOD coding, since indestructibility implies failure of GCH in the general case.) | |
| Jun 12 at 1:32 | comment | added | Elliot Glazer | Does the indestructibility construction give relative consistency of the conjunction of GCH and V=HOD with these large cardinals? | |
| Jun 12 at 1:32 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Jun 12 at 1:25 | comment | added | Joel David Hamkins | My standard lifting-argument cases have been: measurable, strong, partially strong, supercompact, partially supercompact, strongly compact, partially strongly compact, almost huge, huge, rank-to-rank. I've less often worked with extendible, but the rank-to-rank methods often adapt directly. Meanwhile, there is sometimes an issue of the sort identified in my paper on never-indestructibility: link.springer.com/article/10.1007/s00153-015-0458-3 | |
| Jun 12 at 1:20 | comment | added | Elliot Glazer | Thank you for the thorough answer! Can you comment on the case of extendible cardinals? I’m always nervous about those due to their high quantifier complexity. | |
| Jun 12 at 1:15 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Jun 12 at 1:03 | history | answered | Joel David Hamkins | CC BY-SA 4.0 |