Asaf Karagila
I don't have much choice...

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.

By
(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 :)

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

(correction: $\langle \mathbb{P},\mathcal{G},\mathcal{F}_I\rangle$ is a coloopsing symmetric forcing iff .....)

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