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.

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Definitions

The diamond principle $◊$ says that there exists a ◊-sequence, a family of sets $A α \subseteq α$ for $α < ω 1$ such that for any subset $A$ of ω₁ the set of $α$ with $A \cap α = A α$ is stationary in $ω 1$ .

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 $A \cap α \in A α$ and $C \cap α \in A α$ . Another equivalent form states that there exist sets $A α \subseteq α$ for $α < ω 1$ such that for any subset $A$ of $ω 1$ there is at least one infinite $α$ with $A \cap α = A α$ .

More generally, for a given cardinal number $κ$ and a stationary set $S \subseteq κ$ , the statement $◊ S$ (sometimes written $◊(S)$ or $◊ κ (S)$ ) is the statement that there is a sequence $⟨ A α : α \in S ⟩$ such that

each $A α \subseteq α$
for every $A \subseteq κ$ , ${α \in S : A \cap α = A α}$ is stationary in $κ$

The principle $◊ ω 1$ 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 $A \cap α \in A α$ and $C \cap α \in A α$ .

Properties and use

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 $♣ + \neg CH$ , so $◊$ and $♣$ are not equivalent (rather, $♣$ is weaker than $◊$ ).

Matet proved the principle $\diamondsuit _{\kappa }$ equivalent to a property of partitions of $\kappa$ with diagonal intersection of initial segments of the partitions stationary in $\kappa$ .^[1]

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 $S \subseteq κ +$ , $◊ S$ 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.

References

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. PMC 419638. PMID 15131270.
Jensen, R. Björn (1972). "The fine structure of the constructible hierarchy". Annals of Mathematical Logic. 4 (3): 229–308. doi:10.1016/0003-4843(72)90001-0. MR 0309729.
Rinot, Assaf (2011). "Jensen's diamond principle and its relatives". Set theory and its applications. Contemporary Mathematics. Vol. 533. Providence, RI: AMS. pp. 125–156. arXiv:0911.2151. Bibcode:2009arXiv0911.2151R. ISBN 978-0-8218-4812-8. MR 2777747.
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. S2CID 123351674.
Shelah, Saharon (2010). "Diamonds". Proceedings of the American Mathematical Society. 138 (6): 2151–2161. doi:10.1090/S0002-9939-10-10254-8.

Citations

^ P. Matet, "On diamond sequences". Fundamenta Mathematicae vol. 131, iss. 1, pp.35--44 (1988)

This page was last edited on 13 February 2024, at 12:12

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