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The classical Namba forcing collapses $\omega_2$ to have cofinality $\omega$ while preserving the cardinal $\omega_1$. Higher analogs were constructed to (under some additional hypotheses) give a cardinal $\omega_n$ cofinality $\omega$ while preserving $\omega_k$ for $k<n$.

However, i am interested in a different generalisation which collapses to $\omega_1$ instead of $\omega$. So: Does there (consistently) exist a forcing notion that forces $\omega_3$ to have cofinality $\omega_1$ while preserving $\omega_2$?

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    $\begingroup$ Since this will violate covering, it will require a stronger hypothesis than just ZFC. $\endgroup$
    – Joel David Hamkins
    Commented Feb 26 at 17:47
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    $\begingroup$ It is possible with the help of a Woodin cardinal and a stationary tower forcing $\mathbb{P}_\delta$. (You may find a proof from Larson's book about the stationary tower.) $\endgroup$
    – Hanul Jeon
    Commented Feb 26 at 20:39
  • $\begingroup$ You are right, it is example 2.4.4. If you add your comment as an answer, id happily accept it. $\endgroup$
    – Hannes Jakob
    Commented Feb 27 at 14:05
  • $\begingroup$ I may add an answer to your question, but I think an answer for a relevant question is available on MSE, which is better. $\endgroup$
    – Hanul Jeon
    Commented Feb 28 at 17:53

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