Timeline for Are larger large cardinals less expressible?
Current License: CC BY-SA 3.0
11 events
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| Sep 11, 2013 at 14:45 | comment | added | Joel David Hamkins | I agree with that, but I take the question to be about $\Pi_1$ in $V$, rather than $\Pi^1_1$ over $V_\kappa$. | |
| Sep 11, 2013 at 14:44 | comment | added | Monroe Eskew | Sorry, I meant $\Pi^1_1$. Witnessing hugeness measures can be found in $V_\kappa$. | |
| Sep 11, 2013 at 14:18 | comment | added | Joel David Hamkins | @MonroeEskew, is that really $\Pi_1$? It looks at least $\Pi_2$ to me, since to say that $\delta$ is huge is $\Sigma_2$. | |
| Sep 11, 2013 at 14:15 | comment | added | Monroe Eskew | I don't know if you'd count this as a large cardinal, but a very strong $\Pi_1$ property of $\kappa$ is there are stationary many huge cardinals below $\kappa$. | |
| Sep 11, 2013 at 13:13 | comment | added | user36136 | So there is no direct relevance between largeness of a large cardinal and its first order expressibility. Do you have any information about this kind of relevance in a different sense? | |
| Sep 11, 2013 at 13:06 | comment | added | Joel David Hamkins | I suppose one could take a look through Cantor's upper attic cantorsattic.info/Upper_attic. It seems that $\Pi_1$ doesn't get you very far past Mahloness. Of course, you can get to hyper-Mahlo and somewhat beyond, but I'm not sure that anything larger is $\Pi_1$. Meanwhile, consistency assertions are $\Pi^0_1$, such as $\text{Con}(\text{ZFC}+\text{proper class of supercompact cardinals})$, and these can have very high consistency strength. So if these count as "large cardinal properties", then we shouldn't expect a largest one. | |
| Sep 11, 2013 at 13:01 | vote | accept | CommunityBot | ||
| Sep 11, 2013 at 13:01 | comment | added | user36136 | Thanks for your elegant answers. What about question (3)? | |
| Sep 11, 2013 at 12:59 | vote | accept | CommunityBot | ||
| Sep 11, 2013 at 13:01 | |||||
| Sep 11, 2013 at 12:18 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
added 317 characters in body
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| Sep 11, 2013 at 10:49 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |