# To Colloops a cardinal

There are 2 comments on this post.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).

## There are 2 comments on this post.

(Oct 14 2014, 13:21)

Hehe, I like the funny name "coloopsing"! How do you mean that this is a choice-y way of thinking about it, this is a symmetric forcing construction after all. The paragraph about the Feferman-Levy model seems to avoid spelling out that one needs an automorphism group with a normal filter ;)

About a characterisation, I think it's enough to say that such a symmetric forcing would be (perhaps embeddable to) a product of Levy collapses of a sequence cofinal to the particular cardinal, with a projectable symmetry generator that has projections at all levels of the cofinal sequence but excludes the cofinal sequence itself: \(\langle \mathbb{P},\mathcal{G},\mathcal{F}_I \rangle \) is a coloopsing forcing for a cardinal \(\kappa\) with respect to a regular cardinal \(\eta\leq \text{cof}\kappa\) iff for some sequence \(\{ \alpha_\xi\ ; \ \xi<\text{cof}\kappa\}\) that is cofinal in \(\kappa\),

\(\mathbb{P}\)is (densely or completely embeddable to, or equal to, either way) \(\{ p: \text{cof}\kappa \times \eta \rightharpoonup\kappa\ ; \ |p|<\eta \land \forall(\xi,\beta)\in\text{dom}(p)(p(\xi,\beta)<\alpha_\xi)\}\), \(\mathcal{G}\) is the full permutation group of \(\eta\), and \(\mathcal{F}_I\) is the normal filter generated by the \(\mathcal{G}\)-symmetry generator \(I:=\{ E_\xi\ ; \ \xi<\text{cof}\kappa\}\), where for \(\xi<\text{cof}\kappa\), \(E_\xi:=\{p\cap(\xi\times\eta\times\alpha_\xi)\ ; \ p\in\mathbb{P}\}\).

For a singular \(\kappa\) this gives a Feferman-Levy type of symmetric model, and for a regular \(\kappa\) this is equivalent to a Jech type of symmetric model.

But since you say you don't expect a characterisation, I'm hoping to see an example of a coloops that doesn't fit the above description :)

(Oct 14 2014, 13:25 In reply to Ioanna M. Dimitriou)

(correction: \(\langle \mathbb{P},\mathcal{G},\mathcal{F}_I\rangle\) is a coloopsing

symmetricforcing iff .....)