mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, and particularly in
axiomatic set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
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 L, is a particular Class (set theory), class of Set (mathematics), sets that can be described entirely in terms of simpler sets. L is the un ...
() and that implies the
continuum hypothesis
In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states:
Or equivalently:
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this ...
. 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''. The axiom, first investigated by Kurt Gödel, is inconsistent with the pr ...
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, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...
and a
stationary set
In mathematics, specifically set theory and model theory, a stationary set is a set that is not too small in the sense that it intersects all club sets and is analogous to a set of non-zero measure in measure theory. There are at least three closel ...
, 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 cal ...
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 Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
Shelah Shelah may refer to:
* Shelah (son of Judah), a son of Judah according to the Bible
* Shelah (name), a Hebrew personal name
* Shlach, the 37th weekly Torah portion (parshah) in the annual Jewish cycle of Torah reading
* Salih, a prophet described i ...
gave models of , so and are not equivalent (rather, is weaker than ).
Matet proved the principle equivalent to a property of partitions of with diagonal intersection of initial segments of the partitions stationary in .P. Matet, On diamond sequences . Fundamenta Mathematicae vol. 131, iss. 1, pp.35--44 (1988)
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
...
, 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 "student John Smith is not lazy" is a c ...
constructible universe
In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by L, is a particular Class (set theory), class of Set (mathematics), sets that can be described entirely in terms of simpler sets. L is the un ...
. 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 a ...
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 statem ...