# $GCH$ and special Aronszajn trees

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 ?''.

• 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! – Morteza Azad Oct 24 '16 at 1:38
• 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 – Asaf Karagila Aug 24 '18 at 8:43
• @AsafKaragila Aspero's silde is now avalable Special $\aleph_2$-Aronszajn trees and GCH. I may post an answer to this question later. – Mohammad Golshani Aug 28 '18 at 7:33
• 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. – Asaf Karagila Sep 21 '18 at 23:06

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.
• Torres-Prez. Your answer is not correct, Specker's construction gives a special $\aleph_2$-Aronszajn tree. I'm asking for non-special ones. – Mohammad Golshani Oct 5 '17 at 7:23