Huge cardinals are huge!
There are 2 comments on this post.In a previous post, I gave a humorous classification of large cardinals, dividing them to large large cardinals and small large cardinals, and so on. In particular huge cardinals were classified as large large large large large cardinals. But how large are they? Not surprisingly, very large.
In case you forgot, \(\kappa\) is a huge cardinal if there is an elementary embedding \(j\colon V\to M\), where \(M\) is a transitive class containing all the ordinals, with \(\kappa\) critical, and \(M\) is closed under sequences of length \(j(\kappa)\).
This means that \(V_{j(\kappa)}\subseteq M\), and since \(M\) satisfies that \(j(\kappa)\) is an inaccessible cardinal, this means that \(V\) satisfies that \(j(\kappa)\) is an inaccessible cardinal. And in fact, this proof can be squeezed to get much more than just that (with regards to Joel D. Hamkins).
You may recall, from that previous post, that just inaccessible cardinals are, as the smallest of the large cardinals, small small small large cardinals.
And so we have that a huge cardinal implies the existence of larger inaccessible cardinals. This means that huge cardinals, those large large large large large cardinals, are so large, that they imply the existence of even larger small small small small large cardinals!
And that's large!
There are 2 comments on this post.
(Oct 05 2014, 19:26)
Nice post, Asaf. The property that \(V_{j(\kappa)}\subset M\) is really only using the superstrongness of \(\kappa\), rather than the almost hugeness. Also, it may be interesting to point out that there are much smaller large cardinals that also have the kind of implication in which you are interested. For example, if \(\kappa\) is uplifting (a concept weaker than Mahlo in consistency strength), then the inaccessible cardinals above \(\kappa\) form a proper class.
(Oct 05 2014, 21:12 In reply to Joel David Hamkins)
Thank you Joel.
I am aware that you can get away with much weaker cardinals. But it's part of the joke that huge cardinals are large large large large large cardinals, so large that they imply the existence of small small small large cardinals which are larger than themselves! :-)