Almost-huge cardinals are characterizable in terms of coherent towers of supercompactness measures, with a certain property of the direct limit model (see Kanamori's book). A useful large cardinal hypothesis between almost-huge and huge is the assertion that a cardinal has an almost-hugeness towers of unbounded height. The most natural name for this would be "super-almost-huge," but that sounds too awkward to me. I was thinking "ultracompact" might be good because of its affinity with supercompactness. So my question is, is this a bad choice of terminology for some reason, or did someone already give a name to this hypothesis?
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1$\begingroup$ Sometimes people say that $\kappa$ is super-almost-huge to mean that $\kappa$ is almost huge with arbitrarily large almost-hugeness targets, that is, witnessed by almost hugeness embeddings $j:V\to M$ where $j(\kappa)$ is as large as you like. $\endgroup$– Joel David HamkinsCommented Oct 23, 2013 at 2:31
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$\begingroup$ Yes, this is equivalent to the towers being unbddly high. $\endgroup$– Monroe EskewCommented Oct 23, 2013 at 2:34
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$\begingroup$ Ah, I wasn't sure what you meant by that. In this case, I would say that it is fairly standard terminology, and widely understood. $\endgroup$– Joel David HamkinsCommented Oct 23, 2013 at 2:37
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$\begingroup$ Re: "ultracompact" itself, a quick google search reveals no use of the term in set theory. In point-set topology, however, an "ultracompact space" is a space such that given any ultrafilter $U\in\beta\mathbb{N}$ and any sequence $(x_i)_{i\in\mathbb{N}}$, there is some $x\in X$ such that $ \forall O\subseteq X\text{ open, } x\in O\implies \{j\in\mathbb{N}: x_j\in O\}\in U.$ $\endgroup$– Noah SchweberCommented Oct 23, 2013 at 2:37
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I think that the super-almost-huge terminology is fairly widely used with this meaning, so that $\kappa$ is super-almost-huge if for every $\theta$ there is an almost hugeness embedding $j:V\to M$ with critical point $\kappa$ and $j(\kappa)\gt\theta$.
This terminology aligns with the similar terminology for superhuge cardinals and super $n$-huge cardinals, which seems to be fairly established. In addition, there are the super high-jump cardinals, and super-almost-high-jump cardinals, which follow the same idea of having these embeddings for arbitrarily large targets. For this reason I would recommend against introducing a new terminology without very good reason.
(Meanwhile, note that "hypercompact" is already taken, as you can see on Norman Perlmutter's chart of large cardinals near the high-jump cardinals (linked above), although to my knowledge "ultracompact" is still available as a large cardinal name.)
answered Oct 23, 2013 at 3:00
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$\begingroup$ The two last links to the picture boolesrings.org/perlmutter/files/2013/04/ChartForWebSite.jpg are broken. (Maybe somebody knows what post they are taken from and they are still available somewhere online.) However, the picture is available in the Wayback Machine. $\endgroup$ Commented Jul 20, 2019 at 7:53
