2
$\begingroup$

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 $\omega_2$?

$\endgroup$
4
  • 2
    $\begingroup$ Since this will violate covering, it will require a stronger hypothesis than just ZFC. $\endgroup$ Feb 26 at 17:47
  • 2
    $\begingroup$ 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
    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. $\endgroup$ 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
    Feb 28 at 17:53

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.