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December 9

Huge cardinal

Article has Huge cardinals and n-Huge cardinals and almost n-Huge cardinals discussed. Also 0-Huge cardinal is measurable. What is almost 0-Huge cardinal, anything? Grutgrutwhatever (talk) 05:02, 9 December 2021 (UTC)

So per our article, an "almost 0-huge" κ would be the critical point of a nontrivial elementary embedding $j:V\to M$ such that M is closed under $<j^0(\kappa )$-sequences; that is, $<\kappa$-sequences. However, because κ is the critical point of a nontrivial elementary embedding at all, κ must be measurable, and therefore must actually be 0-huge.

It seems the conclusion is that "almost 0-huge" is the same as "0-huge" (that is; measurable).

However, it still seems possible that a particular embedding might be almost 0-huge without being 0-huge. That is, a particular embedding j might satisfy the "almost 0-huge" property without satisfying the "0-huge" property, but some different embedding would still witness that the critical point of the first embedding is 0-huge. Off the top of my head I don't know whether that can happen or not. --Trovatore (talk) 06:25, 9 December 2021 (UTC)