Question. Does $\text{GCH}$ imply the existence of a non-special $\aleph_2$-Aronszajn tree ?

Remark 1. By a result of Jensen, it is consistent that $\text{GCH}$ holds and all $\aleph_1$-Aronszajn trees are special.

Remark 2. The above question is related to the famous question ``does $\text{GCH}$ imply the existence of an $\aleph_2$-Souslin tree ?''.

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    $\begingroup$ Here is a reference to your recent joint paper with Yair Chayut on this topic for further reading of those who might be interested in knowing more! $\endgroup$ – Morteza Azad Oct 24 '16 at 1:38
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    $\begingroup$ Funny. This getting bumped just when David Asperó is about to give a talk about this very topic in Amsterdam at the KNAW 2018 Colloquium... :P $\endgroup$ – Asaf Karagila Aug 24 '18 at 8:43
  • $\begingroup$ @AsafKaragila Aspero's silde is now avalable Special $\aleph_2$-Aronszajn trees and GCH. I may post an answer to this question later. $\endgroup$ – Mohammad Golshani Aug 28 '18 at 7:33
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    $\begingroup$ Mohammad, maybe it's time to point out your new paper on arXiv? The community service bumped this up again, today. I think it knows. $\endgroup$ – Asaf Karagila Sep 21 '18 at 23:06

It turned out the answer to the question is no. David Aspero and I, have proved the following:

Theorem. Assuming the existence of a weakly compact cardinal, there exists a generic extension of the universe in which $GCH$ holds and all $\aleph_2$-Aronszajn trees are special.

See The special Aronszajn tree property at $\aleph_2$ and $GCH$.

Comments and suggestions on the paper are welcome.


Yes, this is a result by Specker: Specker, E.. "Sur un problème de Sikorski." Colloquium Mathematicae 2.1 (1949): 9-12.

  • $\begingroup$ Torres-Prez. Your answer is not correct, Specker's construction gives a special $\aleph_2$-Aronszajn tree. I'm asking for non-special ones. $\endgroup$ – Mohammad Golshani Oct 5 '17 at 7:23
  • $\begingroup$ In fact I think the question is open and quite old. $\endgroup$ – Mohammad Golshani Oct 5 '17 at 7:24

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