15
$\begingroup$

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 $GCH$.

$\endgroup$
11
$\begingroup$

Answer to 1, without CH:

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

Answer to 1, with CH:

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

Answer to 1, with GCH:

  • Gregory, 1976: Mahlo cardinal is a lower bound.
  • Rinot, 2016: Weakly compact is a lower bound.

Answer to 2, with GCH:

  • Rinot, 2016 (building on recent work of Schindler and Steel): AD holds in $L(\mathbb R)$ is a lower bound.
$\endgroup$
  • $\begingroup$ my paper is available here: assafrinot.com/paper/24 $\endgroup$ – saf Apr 4 '16 at 15:59
  • $\begingroup$ Very nice Assaf. I'm still wondering what happens if we assume no $GCH$ type assumptions. $\endgroup$ – Mohammad Golshani Apr 5 '16 at 4:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.