# 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.

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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### 7 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”.

0 likes

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).

1 likes

3. saf says:

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

2 likes

May I know what is the answer?

2 likes

• 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 $L$.

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