Generalizing Feferman - Levy 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 $\beth_{n}$? 
 A: Posting this as an answer at Colin's request. The second paragraph of the question is addressed at this other MO question. 
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:


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*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) 

*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
