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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:
- 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
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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 collapsing preserving a singular and preserving 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)
- 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
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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 collapsing a singular and preserving 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)
- 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
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