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 $\kappa$-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 $\square_\lambda$ entails a $\lambda^+$-Souslin tree for all cardinals $\lambda\ge\aleph_1$. Recently, it was shown that $\square_\lambda$ may be replaced by the weaker principle $\square(\lambda^+)$. Of course, another weakening of $\square_\lambda$ is the principle $\square^*_\lambda$.

Schimmerling’s question indeed asks whether it is consistent with GCH that $\square^*_\lambda$ holds for a singular cardinal $\lambda$, and yet there exist no $\lambda^+$-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 $\lambda$ is a strongly inaccessible cardinal satisfying $2^\lambda=\lambda^+$. If $\mathbb P$ is a $\lambda^+$-cc notion of forcing of size $\le\lambda^+$ that singularizes $\lambda$, then $\mathbb P$ adds a $\lambda^+$-Souslin tree.

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9 Responses to Prikry forcing may add a Souslin tree

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

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  2. Chris says:

    Do you know if this is also true for Prikry or Radin forcing if $2^\lambda >\lambda^+$? 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.

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    • saf says:

      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 $\diamondsuit$-based.
      Once we are done with this project, I would definitely like to think about constructions that are not $\diamondsuit$-based.

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      • Chris says:

        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 $\lambda$ such that the tree property fails at $\lambda^+$ and yet there are no $\lambda^+$-Souslin trees, regardless of cardinal arithmetic assumptions.

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