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Recall Silver's theorem which says that "if GCH holds below $\aleph_{\omega_1}$, then $2^{\aleph_{\omega_1}}=\aleph_{\omega_1+1},$ i.e., it also holds at $\aleph_{\omega_1}$".

Recently, Gitik has proved a similar result for collapses. See his paper Silver type theorems for collapses.

Here I would like to ask two related questions:

Question 1. Suppose that for each limit ordinal $\alpha < \omega_1,$ there exists a special $\aleph_{\alpha+1}$-Aronszajn tree. Does it follow that there is a special $\aleph_{\omega_1+1}$-Aronszajn tree?

The next question is related to tree property.

Question 2. Suppose that for each limit ordinal $\alpha < \omega_1,$ there exists an $\aleph_{\alpha+1}$-Aronszajn tree. Does it follow that there is an $\aleph_{\omega_1+1}$-Aronszajn tree?

Remark. Let me say why a negative consistency result to, say question 2, is not at least trivial. The simplest way to get the tree property at successor of a singular cardinal $\kappa$ with, say, $cf(\kappa)=\omega_1,$ is to have an increasing and cofinal sequence $(\kappa_\xi: \xi < \omega_1)$ of strongly compact cardinals below $\kappa$. But then, for each limit ordinal $\alpha < \omega_1,$ the tree property holds at the successor of $\sup_{\xi<\alpha}\kappa_\xi$ as well. So $\kappa^+$ in not the least successor of a singular cardinal for which the tree property holds. The natural idea is to kill all possible such cases below $\kappa,$ but then it is not clear to me if we preserve tree property at $\kappa^+.$

On the other hand, maybe the arguments similar to those given by Ari Meir Brodsky and Assaf Rinot in Reduced powers of Souslin trees may give a positive answer to the above questions.

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  • $\begingroup$ Is there a reason to expect such theorems? $\endgroup$ – Asaf Karagila Apr 15 '17 at 12:39
  • $\begingroup$ @AsafKaragila I added some explanations. $\endgroup$ – Mohammad Golshani Apr 16 '17 at 5:07

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