Timeline for Name for an intermediate notion between huge and 2-huge
Current License: CC BY-SA 4.0
17 events
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| Oct 13, 2021 at 20:30 | history | edited | LSpice | CC BY-SA 4.0 |
Capitalise title while it's on the front page
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| Oct 13, 2021 at 19:07 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Sep 13, 2021 at 18:53 | answer | added | Arvid Samuelsson | timeline score: 1 | |
| Apr 24, 2015 at 9:09 | comment | added | Monroe Eskew | @ChrisLambie-Hanson, this answers the question but it looks like there is also some inconsistency in terminology. Woodin says "$\kappa_0$ is $\kappa_1$-huge'' to mean there is a huge embedding $j$ with critical point $\kappa_0$ and $j(\kappa_0) = \kappa_1$. I think Joel's terminology sounds the most clear and robust: "crit = x, target = y, closure = z..." | |
| Apr 24, 2015 at 9:05 | comment | added | Chris Lambie-Hanson | In Magidor and Shelah's paper, "The tree property at successors of singular cardinals," a cardinal $\kappa$ is called $\tau$-huge if there is an elementary embedding $j:V \rightarrow M$ with critical point $\kappa$ such that $\kappa < \tau < j(\kappa) < j(\tau)$ and $M^{j(\tau)} \subseteq M$. | |
| Apr 23, 2015 at 16:23 | comment | added | Monroe Eskew | Looks like it doesn't work by strength considerations. Suppose $\kappa$ is $\delta$-huge with closure $\lambda$, and $2^\delta<\lambda$. Then some reflection arguments show that for stationary many $\alpha < \kappa$, $V_\kappa \models \alpha$ is supercompact and huge. | |
| Apr 23, 2015 at 15:23 | comment | added | Joel David Hamkins | If $\kappa<\lambda$, then in my paper on tall cardinals, I proved that $\kappa$ is tall with closure $\lambda$ just in case $\kappa$ is tall and also $\kappa$ is $\lambda$-supercompact. That is, the unified property is simply equivalent to the conjunction of the two separate properties. I couldn't make this work for your property (that is, having $\kappa$ be $\lambda$-supercompact and also huge with target below $\lambda$), but do you know that this doesn't work? | |
| Apr 23, 2015 at 13:15 | comment | added | Joel David Hamkins | I like the fanciful names. But meanwhile, there is a notion "tall with closure $\lambda$", and in analogy with that, you could say that $\kappa$ is 1-huge with closure $\lambda$. For example, $n$-huge with closure $\lambda$ would mean $j:V\to M$ is $n$-huge and also $M^\lambda\subset M$. So the interesting case arises when $j^n(\kappa)<\lambda$. | |
| Apr 23, 2015 at 13:06 | comment | added | Miha Habič | $\lambda$-sesquihuge is interesting, but inevitably someone will want to look at similar things between $n$- and $(n+1)$-huge cardinals and I don't think our Latin forefathers gave us enough prefixes for all of these. | |
| Apr 23, 2015 at 11:25 | comment | added | Asaf Karagila♦ | It looks kinda similar to the high jump cardinals in the paper I linked. Although I haven't sat down to contemplate the exact definition there. | |
| Apr 23, 2015 at 10:19 | comment | added | Monroe Eskew | Actually I like it. $\lambda$-sesquihuge. Your comment was good, phantom commenter. | |
| Apr 23, 2015 at 9:38 | comment | added | Dominic van der Zypen | I suggest $]1,2[$-huge because it states in a clear and short form what you want. | |
| Apr 23, 2015 at 9:37 | comment | added | Asaf Karagila♦ | arxiv.org/abs/1307.7387 might have a term for this cardinal. | |
| Apr 23, 2015 at 9:33 | comment | added | Monroe Eskew | Oy vey. Well at least I want something that talks about $\lambda$, similar to "$\lambda$-supercompact." | |
| Apr 23, 2015 at 9:32 | comment | added | Asaf Karagila♦ | Some suggestions (in case no one named these cardinals before): colossal cardinal, vast cardinal, yonder cardinal, 1.5-huge cardinal, and I can continue perhaps indefinitely. | |
| Apr 23, 2015 at 9:08 | history | edited | Monroe Eskew | CC BY-SA 3.0 |
added 11 characters in body
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| Apr 23, 2015 at 8:46 | history | asked | Monroe Eskew | CC BY-SA 3.0 |