To Colloops a cardinal

This is nothing new, but it’s a choice-y way of thinking about it. Which is really what I enjoy doing.

Definition. Let $V$ be a model of $\ZFC$, and $\PP\in V$ be a notion of forcing. We say that a cardinal $\kappa$ is “colloopsed” by $\PP$ (to $\mu$) if every $V$-generic filter $G$ adds a bijection from $\mu$ onto $\kappa$, but there is an intermediate $N\subseteq V[G]$ satisfying $\ZF$ in which there is no such bijection, but there is one for each $\lambda\lt\kappa$.

This means that $\kappa$ has been collapsed by accident! Oops! Or rather, it collapsed just because the axiom of choice is present. If we take $\PP$ to be the (finite support) product of $\operatorname{Col}(\omega,\omega_n)$, then $\aleph_\omega$ is colloopsed, but not collapsed. Namely, by restricting ourselves to the inner model defined by bounded collapses we can easily show that $\aleph_\omega$ is in fact the new $\aleph_1$. This is the Feferman-Levy model (under the assumption that the ground model satisfied $V=L$ anyway).

So from now on, when you apply a Levy collapse argument to a singular cardinal, you don't collapse it, you colloops it. I wonder if there is a nice characterization of colloopsing forcings. But I don't expect that to happen (a man can dream, though).

Huge cardinals are huge!

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!