# Consistency strength of $\aleph_2$-Souslin hypothesis

Question 1. What is known about the consistency strength of $\aleph_2$-Souslin hypothesis?

Question 2. What is known about the consistency strength of having both $\aleph_2$-Souslin hypotheis and $\aleph_3$-Souslin hypothesis?

Remark 1. By $\kappa$-Souslin hopothesis, I mean there are no $\kappa$-Souslin trees.

Remark 2. By Laver-Shelah, the existence of a weakly compact cardinal implies the consistency of $\aleph_2$-Souslin hypothesis. On the other hand by results of Shelah-Stanly, if we assume some instances of $GCH$+ $\aleph_2$-Souslin hypothesis (having $CH$ is sufficient), then some large cardinals (at least Mahlo) are required. In the above question I do not take care of preserving instances of $GCH$.

• Mitchell and Silver, 1973: Weakly compact is an upper bound.

• Laver and Shelah, 1981: Weakly compact is an upper bound.
• Shelah and Stanley, 1982: Inaccessible is a lower bound.

• Rinot, 2016 (building on recent work of Schindler and Steel): AD holds in $L(\mathbb R)$ is a lower bound.
• Very nice Assaf. I'm still wondering what happens if we assume no $GCH$ type assumptions. – Mohammad Golshani Apr 5 '16 at 4:02