large cardinals | Asaf Karagila

Posts tagged large cardinals

Constructive proof that large cardinals are consistent

Jul 06 2016, 13:30

I am not a Platonist, as I keep pointing out. Existence, even not in mathematics, is relative and confusing to begin with, so I don't pretend to try and understand it in a meaningful way.

However, we have a proof, a constructive proof that large cardinals are consistent. And they exist in an inner model of our universe.

Huge cardinals are huge!

Oct 03 2014, 05:32

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 \(j(\kappa)\).

Ramsey cardinals are large large small large cardinals

Sep 19 2014, 22:03

There is no well defined notion for what is a large cardinal. In some contexts those are inaccessibles, in others those are critical points of elementary embeddings, and sometimes \(\aleph_\omega\) is a large cardinal.

But we can clearly see some various degrees of largeness by how much structure the existence of the cardinal imposes. Inaccessible cardinals prove there is a model for second-order \(\ZFC\), and Ramsey cardinals imply \(V\neq L\). Strongly compact cardinals even imply that \(\forall A(V\neq L[A])\).