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:
- Diamond
- Weak Diamond and the Uniformization Property
- The Souslin Hypothesis and Club Guessing
- Saturation of the Nonstationary Ideal
- Index
- References
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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.
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.
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.
Looks like I solved Question 9. Gotta have some rest; going to sleep…
May I know what is the answer?
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).
Specifically, we get that GCH+$\square(\aleph_2)$ entails the existence of a $\sigma$-complete $\aleph_2$-Souslin tree.
Update: See here.
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Update: The answer to Question 16 is “yes”. See here.