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.

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

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}* :

- 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

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.