# Diamond principle

In mathematics, and particularly in axiomatic set theory, the **diamond principle** ◊ is a combinatorial principle introduced by Ronald Jensen in Jensen (1972) that holds in the constructible universe (*L*) and that implies the continuum hypothesis. Jensen extracted the diamond principle from his proof that the Axiom of constructibility (*V* = *L*) implies the existence of a Suslin tree.

## Definitions[edit]

The diamond principle ◊ says that there exists a **◊-sequence**, in other words sets *A _{α}* ⊆

*α*for

*α*<

*ω*

_{1}such that for any subset

*A*of ω

_{1}the set of

*α*with

*A*∩

*α*=

*A*is stationary in

_{α}*ω*

_{1}.

There are several equivalent forms of the diamond principle. One states that there is a countable collection **A**_{α} of subsets of *α* for each countable ordinal *α* such that for any subset *A* of *ω*_{1} there is a stationary subset *C* of *ω*_{1} such that for all *α* in *C* we have *A* ∩ *α* ∈ **A**_{α} and *C* ∩ *α* ∈ **A**_{α}. Another equivalent form states that there exist sets *A*_{α} ⊆ *α* for *α* < *ω*_{1} such that for any subset *A* of *ω*_{1} there is at least one infinite *α* with *A* ∩ *α* = *A*_{α}.

More generally, for a given cardinal number *κ* and a stationary set *S* ⊆ *κ*, the statement ◊_{S} (sometimes written ◊(*S*) or ◊_{κ}(*S*)) is the statement that there is a sequence ⟨*A _{α}* :

*α*∈

*S*⟩ such that

- each
*A*⊆_{α}*α* - for every
*A*⊆*κ*, {*α*∈*S*:*A*∩*α*=*A*} is stationary in_{α}*κ*

The principle ◊_{ω1} is the same as ◊.

The diamond-plus principle ◊^{+} states that there exists a **◊ ^{+}-sequence**, in other words a countable collection

**A**

_{α}of subsets of

*α*for each countable ordinal α such that for any subset

*A*of

*ω*

_{1}there is a closed unbounded subset

*C*of

*ω*

_{1}such that for all

*α*in

*C*we have

*A*∩

*α*∈

**A**

_{α}and

*C*∩

*α*∈

**A**

_{α}.

## Properties and use[edit]

Jensen (1972) showed that the diamond principle ◊ implies the existence of Suslin trees. He also showed that *V* = *L* implies the diamond-plus principle, which implies the diamond principle, which implies CH. In particular the diamond principle and the diamond-plus principle are both independent of the axioms of ZFC. Also ♣ + CH implies ◊, but Shelah gave models of ♣ + ¬ CH, so ◊ and ♣ are not equivalent (rather, ♣ is weaker than ◊).

The diamond principle ◊ does not imply the existence of a Kurepa tree, but the stronger ◊^{+} principle implies both the ◊ principle and the existence of a Kurepa tree.

Akemann & Weaver (2004) used ◊ to construct a *C**-algebra serving as a counterexample to Naimark's problem.

For all cardinals *κ* and stationary subsets *S* ⊆ *κ*^{+}, ◊_{S} holds in the constructible universe. Shelah (2010) proved that for *κ* > ℵ_{0}, ◊_{κ+}(*S*) follows from 2^{κ} = *κ*^{+} for stationary *S* that do not contain ordinals of cofinality *κ*.

Shelah showed that the diamond principle solves the Whitehead problem by implying that every Whitehead group is free.

## See also[edit]

## References[edit]

- Akemann, Charles; Weaver, Nik (2004). "Consistency of a counterexample to Naimark's problem".
*Proceedings of the National Academy of Sciences*.**101**(20): 7522–7525. arXiv:math.OA/0312135. Bibcode:2004PNAS..101.7522A. doi:10.1073/pnas.0401489101. MR 2057719.CS1 maint: ref=harv (link) - Jensen, R. Björn (1972). "The fine structure of the constructible hierarchy".
*Annals of Mathematical Logic*.**4**: 229–308. doi:10.1016/0003-4843(72)90001-0. MR 0309729.CS1 maint: ref=harv (link) - Rinot, Assaf (2011). "Jensen's diamond principle and its relatives".
*Set theory and its applications*. Contemporary Mathematics.**533**. Providence, RI: AMS. pp. 125–156. arXiv:0911.2151. Bibcode:2009arXiv0911.2151R. ISBN 978-0-8218-4812-8. MR 2777747.CS1 maint: ref=harv (link) - Shelah, Saharon (1974). "Infinite Abelian groups, Whitehead problem and some constructions".
*Israel Journal of Mathematics*.**18**(3): 243–256. doi:10.1007/BF02757281. MR 0357114.CS1 maint: ref=harv (link) - Shelah, Saharon (2010). "Diamonds".
*Proceedings of the American Mathematical Society*.**138**: 2151–2161. doi:10.1090/S0002-9939-10-10254-8.CS1 maint: ref=harv (link)