Whitehead problem
Encyclopedia
In group theory
, a branch of abstract algebra
, the Whitehead problem is the following question:
Shelah (1974) proved that Whitehead's problem is undecidable
within standard ZFC set theory.
whose kernel
is isomorphic to the group of integer
s Z, then there exists a group homomorphism
g : A → B with fg = idA
. Abelian groups satisfying this condition are sometimes called Whitehead groups, so Whitehead's problem asks: is every Whitehead group free?
Caution: The converse of Whitehead's problem, namely that every free abelian group is Whitehead, is a well known group-theoretical fact. Some authors call Whitehead group only a non-free group A satisfying Ext1(A, Z) = 0. Whitehead's problem then asks: do Whitehead groups exist?
. More precisely, he showed that:
Since the consistency
of ZFC implies the consistency of either of the following:
Whitehead's problem cannot be resolved in ZFC.
, motivated by the second Cousin problem, first posed the problem in the 1950s. answered the question in the affirmative for countable groups. Progress for larger groups was slow, and the problem was considered an important one in algebra
for some years.
Shelah's result was completely unexpected. While the existence of undecidable statements had been known since Gödel's incompleteness theorem of 1931, previous examples of undecidable statements (such as the continuum hypothesis
) had all been in pure set theory
. The Whitehead problem was the first purely algebraic problem to be proved undecidable.
later showed that the Whitehead problem remains undecidable even if one assumes the Continuum hypothesis. The Whitehead conjecture is true if all sets are constructible
. That this and other statements about uncountable abelian groups are provably independent of ZFC shows that the theory of such groups is very sensitive to the assumed underlying set theory
.
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, a branch of abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, the Whitehead problem is the following question:
- Is every abelian groupAbelian groupIn abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
A with ExtExt functorIn mathematics, the Ext functors of homological algebra are derived functors of Hom functors. They were first used in algebraic topology, but are common in many areas of mathematics.- Definition and computation :...
1(A, Z) = 0 a free abelian groupFree abelian groupIn abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...
?
Shelah (1974) proved that Whitehead's problem is undecidable
Independence (mathematical logic)
In mathematical logic, independence refers to the unprovability of a sentence from other sentences.A sentence σ is independent of a given first-order theory T if T neither proves nor refutes σ; that is, it is impossible to prove σ from T, and it is also impossible to prove from T that...
within standard ZFC set theory.
Refinement
The condition Ext1(A, Z) = 0 can be equivalently formulated as follows: whenever B is an abelian group and f : B → A is a surjective group homomorphismGroup homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
whose kernel
Kernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
is isomorphic to the group of integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s Z, then there exists a group homomorphism
Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
g : A → B with fg = idA
Identity function
In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...
. Abelian groups satisfying this condition are sometimes called Whitehead groups, so Whitehead's problem asks: is every Whitehead group free?
Caution: The converse of Whitehead's problem, namely that every free abelian group is Whitehead, is a well known group-theoretical fact. Some authors call Whitehead group only a non-free group A satisfying Ext1(A, Z) = 0. Whitehead's problem then asks: do Whitehead groups exist?
Shelah's proof
showed that, given the canonical ZFC axiom system, the problem is independent of the usual axioms of set theoryIndependence (mathematical logic)
In mathematical logic, independence refers to the unprovability of a sentence from other sentences.A sentence σ is independent of a given first-order theory T if T neither proves nor refutes σ; that is, it is impossible to prove σ from T, and it is also impossible to prove from T that...
. More precisely, he showed that:
- If every set is constructibleAxiom of constructibilityThe 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 constructible universe, respectively.- Implications :The axiom of...
, then every Whitehead group is free; - If Martin's axiomMartin's axiomIn the mathematical field of set theory, Martin's axiom, introduced by , is a statement which is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, so certainly consistent with ZFC, but is also known to be consistent with ZF + ¬ CH...
and the negation of the continuum hypothesisContinuum hypothesisIn 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...
both hold, then there is a non-free Whitehead groupWhitehead groupWhitehead group in mathematics may mean:* A group W with Ext=0; see Whitehead problem* For a ring, the Whitehead group Wh of a ring A, equal to K_1...
.
Since the consistency
Consistency
Consistency can refer to:* Consistency , the psychological need to be consistent with prior acts and statements* "Consistency", an 1887 speech by Mark Twain...
of ZFC implies the consistency of either of the following:
- The axiom of constructibilityAxiom of constructibilityThe 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 constructible universe, respectively.- Implications :The axiom of...
(which asserts that all sets are constructible); - Martin's axiomMartin's axiomIn the mathematical field of set theory, Martin's axiom, introduced by , is a statement which is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, so certainly consistent with ZFC, but is also known to be consistent with ZF + ¬ CH...
plus the negation of the continuum hypothesisContinuum hypothesisIn 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...
,
Whitehead's problem cannot be resolved in ZFC.
Discussion
J. H. C. WhiteheadJ. H. C. Whitehead
John Henry Constantine Whitehead FRS , known as Henry, was a British mathematician and was one of the founders of homotopy theory. He was born in Chennai , in India, and died in Princeton, New Jersey, in 1960....
, motivated by the second Cousin problem, first posed the problem in the 1950s. answered the question in the affirmative for countable groups. Progress for larger groups was slow, and the problem was considered an important one in algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
for some years.
Shelah's result was completely unexpected. While the existence of undecidable statements had been known since Gödel's incompleteness theorem of 1931, previous examples of undecidable statements (such as 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...
) had all been in pure set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
. The Whitehead problem was the first purely algebraic problem to be proved undecidable.
later showed that the Whitehead problem remains undecidable even if one assumes the Continuum hypothesis. The Whitehead conjecture is true if all sets are constructible
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"...
. That this and other statements about uncountable abelian groups are provably independent of ZFC shows that the theory of such groups is very sensitive to the assumed underlying set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
.
See also
- Free abelian groupFree abelian groupIn abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...
- Whitehead groupWhitehead groupWhitehead group in mathematics may mean:* A group W with Ext=0; see Whitehead problem* For a ring, the Whitehead group Wh of a ring A, equal to K_1...
- List of statements undecidable in ZFC
- Statements true if all sets are constructibleStatements true in LHere is a list of propositions that hold in the constructible universe :* The generalized continuum hypothesis and as a consequence** The axiom of choice* Diamondsuit** Clubsuit* Global square* The existence of morasses...