Mathematics desk
< December 8	<< Nov \| December \| Jan >>	Current desk >

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

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)[reply]

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)[reply]