In
mathematics, and particularly in
axiomatic set theory, the diamond principle is a
combinatorial principle In proving results in combinatorics several useful combinatorial rules or combinatorial principles are commonly recognized and used.
The rule of sum, rule of product, and inclusion–exclusion principle are often used for enumerative purposes. B ...
introduced by
Ronald Jensen
Ronald Björn Jensen (born April 1, 1936) is an American mathematician who lives in Germany, primarily known for his work in mathematical logic and set theory.
Career
Jensen completed a BA in economics at American University in 1959, and a Ph.D. ...
in that holds in the
constructible universe
In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. is the union of the constructible hierarchy . It w ...
() and that implies the
continuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
. Jensen extracted the diamond principle from his proof that the
axiom of constructibility
The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. The axiom is usually written as ''V'' = ''L'', where ''V'' and ''L'' denote the von Neumann universe and the construc ...
() implies the existence of a
Suslin tree.
Definitions
The diamond principle says that there exists a , a family of 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
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. T ...
and a
stationary set , the statement (sometimes written or ) is the statement that there is a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
such that
* each
* for every , is stationary in
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 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
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independ ...
of the axioms of ZFC. Also implies , but
Shelah gave models of , so and are not equivalent (rather, is weaker than ).
The diamond principle does not imply the existence of a
Kurepa tree
In set theory, a Kurepa tree is a tree (''T'', <) of height ω1, each of whose levels is at most countable, and has at ...
, but the stronger principle implies both the principle and the existence of a Kurepa tree.
used to construct a
-algebra serving as a
counterexample
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...
to
Naimark's problem Naimark's problem is a question in functional analysis asked by . 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 (not necess ...
.
For all cardinals and
stationary subsets , holds in the
constructible universe
In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. is the union of the constructible hierarchy . It w ...
. proved that for , follows from for stationary that do not contain ordinals of cofinality .
Shelah showed that the diamond principle solves the
Whitehead problem
In group theory, a branch of abstract algebra, the Whitehead problem is the following question:
Saharon Shelah proved that Whitehead's problem is independent of ZFC, the standard axioms of set theory.
Refinement
Assume that ''A'' is an abel ...
by implying that every
Whitehead group is free.
See also
*
List of statements independent of ZFC
The mathematical statements discussed below are independent of ZFC (the canonical axiomatic set theory of contemporary mathematics, consisting of the Zermelo–Fraenkel axioms plus the axiom of choice), assuming that ZFC is consistent. A stateme ...
*
Statements true in
References
*
*
*
*
*
{{refend
Set theory
Mathematical principles
Independence results
Constructible universe