The Feferman - Levy model makes $\aleph\_1$ singular by a cardinal collapse $\aleph\_1 = \aleph\_{\omega}^L$. Unless I've got something wrong, the same thing would work to make any well-orderable cardinal $\alpha$ cofinal in its well-ordered cardinal successor. Is that right?


The Feferman -Levy model also makes the continuum a countable union of countable sets. Does that generalize to Beth numbers, in the sense of successive power sets starting with $\omega$? For each finite $n$, are there models where $\beth\_{n+1}$ is a union of $\beth\_{n}$ many sets each smaller than or the same size as $\beth\_{n}$? 



Generalizing Feferman - Levy
Posting this as an answer at Colin's request. The second paragraph of the question is addressed at this other [MO question](https://mathoverflow.net/questions/109680/does-zf-bound-countable-unions-of-countable-sets). 


The answer to the question in the first paragraph is delicate, it depends on how much of the ground model we decide to preserve: $\alpha$ and $\alpha^+$? $\rm{cof}(\alpha)$ and $\alpha^+$? Only $\alpha^+$? The issue is that a straight generalization of Feferman-Lévy must fail just based on consistency strength considerations, because Jensen's covering lemma gives us that preserving a singular and collapsing its successor requires large cardinals (even in $\mathsf{ZF}$). For the covering lemma, see:


* Devlin, Keith I.; Jensen, Ronald B. *Marginalia to a theorem of Silver*. In **$\models$ISILC Logic Conference (Proc. Internat. Summer Inst. and Logic Colloq., Kiel, 1974)**, G. H. Müller, A. Oberschelp, and K. Potthoff, eds., pp. 115–142. Lecture Notes in Math., Vol. 499, Springer, Berlin, 1975. [MR0480036 (58 #235)](http://www.ams.org/mathscinet-getitem?mr=480036)
* Mitchell, William J. *The covering lemma*. In **Handbook of set theory. Vols. 1, 2, 3**, Kanamori, Foreman, eds., pp. 1497–1594, Springer, Dordrecht, 2010. [MR2768697](http://www.ams.org/mathscinet-getitem?mr=2768697)


