Prikry forcing may add a Souslin tree

A celebrated theorem of Shelah states that adding a Cohen real introduces a Souslin tree. Are there any other examples of notions of forcing that add a $κ$ -Souslin tree? and why is this of interest?

My motivation comes from a question of Schimmerling, which I shall now motivate and state.
Recall that Jensen proved that GCH together with the square principle $_{λ}$ entails a $λ^{+}$ -Souslin tree for all cardinals $λ \geq ℵ_{1}$ . Recently, it was shown that $_{λ}$ may be replaced by the weaker principle $(λ^{+})$ . Of course, another weakening of $_{λ}$ is the principle $_{λ}^{*}$ .

Schimmerling’s question indeed asks whether it is consistent with GCH that $_{λ}^{*}$ holds for a singular cardinal $λ$ , and yet there exist no $λ^{+}$ -Souslin trees.
The first line of attacks that comes to mind here would involve Prikry/Magidor/Radin forcing to singularize a former large cardinal (e.g., this paper).
In this post, we announce a (corollary to a) theorem from an upcoming paper with Brodsky, showing that this line of attacks is a no-go.

Theorem. Suppose that $λ$ is a strongly inaccessible cardinal satisfying $2^{λ} = λ^{+}$ . If $P$ is a $λ^{+}$ -cc notion of forcing of size $\leq λ^{+}$ that singularizes $λ$ , then $P$ adds a $λ^{+}$ -Souslin tree.

11 Responses to Prikry forcing may add a Souslin tree

Asaf Karagila says:

June 12, 2016 at 2:20 pm

I think that this adds nicely to the paper Yair and I wrote. Forcings that singularize cardinals do nasty things.

- saf says:
  
  June 12, 2016 at 2:58 pm
  
  Souslin trees are of diamond-quality; quite far from being considered nasty here :).
  
  - Asaf Karagila says:
    
    June 12, 2016 at 4:40 pm
    
    Well… not if you are trying to prove the consistency of weak squares without Suslin trees at a singular cardinal.
    
    - saf says:
      
      June 12, 2016 at 4:50 pm
      
      That would be quite a depressing model.
      
      - Asaf Karagila says:
        
        June 12, 2016 at 6:37 pm
        
        In continuation of my recent post about my experience of Norwich, this conversation should continue only after the third beer has been served.
Chris says:

June 13, 2016 at 3:07 am

Do you know if this is also true for Prikry or Radin forcing if $2^{λ} > λ^{+}$ ? This is the situation in the Cummings, et al. paper you linked to and seems like it would naturally arise in attempts to answer Schimmerling’s question in a similar way.

- saf says:
  
  June 13, 2016 at 3:14 am
  
  Indeed, that’s the situation in the Cummings et al paper, but Schimmerling asks about GCH. About our argument: all of the Souslin trees constructions in the “Microscopic approach” papers are $d i a m o n d s u i t$ -based.
  Once we are done with this project, I would definitely like to think about constructions that are not $d i a m o n d s u i t$ -based.
  
  - Chris says:
    
    June 13, 2016 at 3:28 am
    
    Right, I missed the inclusion of GCH in Schimmerling’s question. As far as I know, it’s already an interesting and open question whether there can be a singular cardinal $λ$ such that the tree property fails at $λ^{+}$ and yet there are no $l a m b d a^{+}$ -Souslin trees, regardless of cardinal arithmetic assumptions.
    
    - saf says:
      
      June 13, 2016 at 3:32 am
      
      I agree!
      
Mohammad says:

February 14, 2018 at 8:07 am

Does your result says anything when $2^{λ} > λ^{+}$ ?

- saf says:
  
  February 14, 2018 at 8:59 am
  
  It does not, because the proof goes through the proxy principle $P (λ^{+}, \dots)$ that asserts, among other things, that $♢ (λ^{+})$ holds.

Prikry forcing may add a Souslin tree

11 Responses to Prikry forcing may add a Souslin tree

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