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 symmetric forcing iff .....)