Jensen’s diamond principle and its relatives

This is chapter 6 in the book Set Theory and Its Applications (ISBN: 0821848127).

Abstract: We survey some recent results on the validity of Jensen’s diamond principle at successor cardinals. We also discuss weakening of this principle such as club guessing, and anti-diamond principles such as uniformization.

A collection of open problems is included.

Table of Contents:

  1. Diamond
  2. Weak Diamond and the Uniformization Property
  3. The Souslin Hypothesis and Club Guessing
  4. Saturation of the Nonstationary Ideal
  5. Index
  6. References


Citation information:

A. Rinot, Jensen’s diamond principle and its relatives, Set Theory and Its Applications, Contemp. Math., 533: 125-156, Amer. Math. Soc., Providence, RI, 2011.

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8 Responses to Jensen’s diamond principle and its relatives

  1. saf says:

    Update: David Aspero announced a negative answer to Question 13 from our list. The title of his paper is “The consistency of a club-guessing failure at the successor of a regular cardinal”.

    Update: Aspero’s paper is in here.


  2. saf says:

    In her Oberwolfach talk, Heike Mildenberger just announced a negative answer to Question 17! This is joint work with Shelah (the paper # is 988).

    Update: The proof contained a gap, so that the problem remains open.


  3. saf says:

    Looks like I solved Question 9. Gotta have some rest; going to sleep…


  4. Mohammad says:

    May I know what is the answer?


    • saf says:

      the answer is “yes”. In particular, if GCH holds, and there are no $\aleph_2$-Souslin trees, then $\aleph_2$ is weakly compact in $L$.

      (though, I am still proofreading).


      • saf says:

        Specifically, we get that GCH+$\square(\aleph_2)$ entails the existence of a $\sigma$-complete $\aleph_2$-Souslin tree.

        Update: See here.


  5. Pingback: Higher Souslin trees and the GCH, revisited | Assaf Rinot

  6. saf says:

    Update: The answer to Question 16 is “yes”. See here.


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