Diamond principle


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

Definitions

The diamond principle says that there exists a , in other words sets for such that for any subset of ω1 the set of with is stationary in.
There are several equivalent forms of the diamond principle. One states that there is a countable collection of subsets of for each countable ordinal such that for any subset of there is a stationary subset of such that for all in we have and. Another equivalent form states that there exist sets for such that for any subset of there is at least one infinite with.
More generally, for a given cardinal number and a stationary set, the statement is the statement that there is a sequence such that
The principle is the same as.
The diamond-plus principle states that there exists a -sequence, in other words a countable collection of subsets of for each countable ordinal α such that for any subset of there is a closed unbounded subset of such that for all in we have and.

Properties and use

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 implies, but Shelah gave models of so and are not equivalent.
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.
used to construct a -algebra serving as a counterexample to Naimark's problem.
For all cardinals and stationary subsets, holds in the constructible universe. proved that for, follows from for stationary that do not contain ordinals of cofinality.
Shelah showed that the diamond principle solves the Whitehead problem by implying that every Whitehead group is free.