Wikipedia:Reference desk/Archives/Mathematics/2021 December 9

= 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" &kappa; 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 &kappa; is the critical point of a nontrivial elementary embedding at all, &kappa; 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)