Diamondsuit
Encyclopedia
In mathematics
, and particularly in axiomatic set theory, the diamond principle ◊ is a combinatorial principle introduced by that holds in the constructible universe
and that implies the continuum hypothesis
. Jensen extracted the diamond principle from his proof that V=L implies the existence of a Suslin tree
.
More generally, for a given cardinal number
and a stationary set
, the statement ◊S (sometimes written ◊(S) or ◊κ(S)) is the statement that there is a sequence
such that
The principle ◊ω1 is the same as ◊.
s.
He also showed that ◊ implies the CH
. Also ♣
+ CH implies ◊, but Shelah
gave models of ♣ + ¬ CH, so ◊ and ♣ are not equivalent (rather, ♣ is weaker than ◊).
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
. Recently Shelah
proved that for κ>ℵ0, ◊κ+ follows from
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, and particularly in axiomatic set theory, the diamond principle ◊ is a combinatorial principle introduced by that holds in the constructible universe
Constructible universe
In mathematics, the constructible universe , denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis"...
and that implies the continuum hypothesis
Continuum hypothesis
In mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...
. Jensen extracted the diamond principle from his proof that V=L implies the existence of a Suslin tree
Suslin tree
In mathematics, a Suslin tree is a tree of height ω1 such thatevery branch and every antichain is at most countable. Every Suslin tree is an Aronszajn tree....
.
Definition
The diamond principle ◊ says that there exists a ◊-sequence, in other words sets Aα⊆α for α<ω1 such that for any subset A of ω1 the set of α with A∩α = Aα is stationary in ω1.More generally, for a given cardinal number
Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
and a stationary setStationary set
In mathematics, particularly in set theory and model theory, there are at least three notions of stationary set:-Classical notion:If \kappa \, is a cardinal of uncountable cofinality, S \subseteq \kappa \,, and S \, intersects every club set in \kappa \,, then S \, is called a stationary set....
, the statement ◊S (sometimes written ◊(S) or ◊κ(S)) is the statement that there is a sequenceSequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
such that- each
- for every
is stationary in 
The principle ◊ω1 is the same as ◊.
Properties and use
showed that the diamond principle ◊ implies the existence of Suslin treeSuslin tree
In mathematics, a Suslin tree is a tree of height ω1 such thatevery branch and every antichain is at most countable. Every Suslin tree is an Aronszajn tree....
s.
He also showed that ◊ implies the CH
Continuum hypothesis
In mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...
. Also ♣
Clubsuit
In mathematics, and particularly in axiomatic set theory, ♣S is a family of combinatorial principles that are weaker version of the corresponding ◊S; it was introduced in 1975 by A...
+ CH implies ◊, but Shelah
Saharon Shelah
Saharon Shelah is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey.-Biography:...
gave models of ♣ + ¬ CH, so ◊ and ♣ are not equivalent (rather, ♣ is weaker than ◊).
used ◊ to construct a C*-algebra serving as a counterexample
Counterexample
In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule. For example, consider the proposition "all students are lazy"....
to Naimark's problem
Naimark's problem
Naimark's problem is a question in functional analysis. It asks whether every C*-algebra that has only one irreducible representation up to unitary equivalence is isomorphic to the algebra of compact operators on some Hilbert space....
.
For all cardinals κ and stationary subsets S⊆κ+, ◊S holds in the constructible universe
Constructible universe
In mathematics, the constructible universe , denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis"...
. Recently Shelah
Saharon Shelah
Saharon Shelah is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey.-Biography:...
proved that for κ>ℵ0, ◊κ+ follows from
.