Ramsey cardinals are large large small large cardinals

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 \(\forall A(V\neq L[A])\).

So we can humorously classify those notions. Large cardinals for us will be from the start regular limit cardinals.

We begin with large large cardinals, those are critical points of elementary embeddings (into transitive classes, of course). The first measurable is a small large large cardinal, and a strongly compact cardinal is a large large large cardinal. Woodin cardinals lie somewhere between (although they are not quite critical points, since the first Woodin is not weakly compact), and those have quite a rich structure so we can say that those are large small large large cardinals, and if we examine the higher levels of Cantor's Attic, then supercompacts are small large large large cardinals, whereas extendible cardinals are considered large large large large cardinals, and those superhuge cardinals are large large large large large cardinals.

On the other side of the scale are small large cardinals, can be divided to small small large cardinals which are compatible with \(V=L\) and have little structural consequences, so inaccessible cardinals are small small small large cardinals. Remarkable, subtle and ineffable cardinals are large small small large cardinals being closer to the point of enforcing \(V\neq L\). And weakly compact cardinals along with the indescribable cardinals are those that make the bulk middle of the small small large cardinals.

Finally we arrive to the gap between \(0^\#\) and measurable cardinals, there lie the large small large cardinals. From \(\omega_1\)-Erdős cardinals which are small large small large cardinals, to the titular Ramsey cardinals which are large large small large cardinals.