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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