List of statements undecidable in ZFC
Encyclopedia
The mathematical
statements discussed below are undecidable
in ZFC (the Zermelo–Fraenkel axioms plus the axiom of choice, the canonical axiomatic set theory of contemporary mathematics), assuming that ZFC is consistent.
proved the first ZFC undecidability result, namely that the consistency of ZFC itself was undecidable in ZFC.
The axiom of constructibility
(V=L, all sets in the universe are constructible
) implies the generalized continuum hypothesis (which states that ℵn = ℶn for every ordinal n) and the combinatorial statement ◊
, which both imply the continuum hypothesis
(which states that ℵ1
= ℶ1
). All these statements are independent of ZFC, as shown by Paul Cohen
and Kurt Gödel
.
Martin's axiom
together with the negation of the continuum hypothesis
is undecidable in ZFC.
Assuming that ZFC is consistent, the existence of large cardinal numbers, such as inaccessible cardinal
s, Mahlo cardinal
s etc., cannot be proved in ZFC. On the other hand, few working set theorists expect their existence to be disproved.
Set theory of the real line
There are many cardinal invariants of the real line, connected with measure theory and statements related to the Baire category theorem
whose exact values are independent of ZFC (in a stronger sense than that the continuum hypothesis is in ZFC. While nontrivial relations can be proved between them, most cardinal invariants can be any regular cardinal
between
and 
). This is a major area of study in set theoretic real analysis. Martin's axiom has a tendency to set most interesting cardinal invariants equal to 
.
Order theory
The answer to Suslin's problem
is independent of ZFC. The diamond principle ◊ proves the existence of a Suslin line, while Martin's axiom
+ the negation of the continuum hypothesis proves that no Suslin line exists.
Existence of Kurepa trees is independent of ZFC.
Group theory
In 1973, Saharon Shelah
showed that the Whitehead problem
("is every abelian group
A with Ext1(A, Z) = 0 a free abelian group?") is independent of ZFC. A group with Ext1(A, Z) = 0 which is not free abelian is called a Whitehead group; Martin's Axiom
+ the negation of the continuum hypothesis proves the existence of a Whitehead group, while the axiom of constructibility
proves that no Whitehead group exists.
for positive functions, where the function is no longer assumed to be measurable but merely that the two iterated integrals are well defined and exist, is independent of ZFC. On the one hand, the continuum hypothesis
implies that there exists a function on the unit square whose iterated integrals are not equal — the function is simply the indicator function of an ordering of [0, 1] equivalent to a well ordering of the cardinal
. A similar example can be constructed using Martin's axiom
. On the other hand, the consistency of the strong Fubini theorem was first shown by Friedman
. It can also be deduced from a variant of Freiling's axiom of symmetry
.
Functional analysis
Garth Dales and Robert M. Solovay
proved in 1976 that Kaplansky's conjecture, namely that there exists a discontinuous homomorphism
from the Banach algebra
C(X) (where X is some infinite compact
Hausdorff space
) into any other Banach algebra, was independent of ZFC. Assuming the continuum hypothesis
enables proving that for any infinite X there exists such a homomorphism into any Banach algebra.
Charles Akemann and Nik Weaver showed in 2003 that the statement "there exists a counterexample to Naimark's problem
which is generated by
, elements" is independent of ZFC.
Miroslav Bačák and Petr Hájek
proved in 2008 that the statement "every Asplund space of density character
has a renorming with the Mazur intersection property" is independent of ZFC. The result is shown using Martin's maximum
axiom, while Mar Jiménez and José Pedro Moreno (1997) had presented a counterexample under the continuum hypothesis
.
As shown by Ilijas Farah and N. Christopher Phillips, the existence of outer automorphisms of the Calkin algebra
depends on set theoretic assumptions beyond ZFC.
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...
statements discussed below are 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...
in ZFC (the Zermelo–Fraenkel axioms plus the axiom of choice, the canonical axiomatic set theory of contemporary mathematics), assuming that ZFC is consistent.
Axiomatic set theory
In 1931, Kurt GödelKurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
proved the first ZFC undecidability result, namely that the consistency of ZFC itself was undecidable in ZFC.
The axiom of constructibility
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 constructible universe, respectively.- Implications :The axiom of...
(V=L, all sets in the universe 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"...
) implies the generalized continuum hypothesis (which states that ℵn = ℶn for every ordinal n) and the combinatorial statement ◊
Diamondsuit
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...
, which both imply 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...
(which states that ℵ1
Aleph number
In set theory, a discipline within mathematics, the aleph numbers are a sequence of numbers used to represent the cardinality of infinite sets. They are named after the symbol used to denote them, the Hebrew letter aleph...
= ℶ1
Beth number
In mathematics, the infinite cardinal numbers are represented by the Hebrew letter \aleph indexed with a subscript that runs over the ordinal numbers...
). All these statements are independent of ZFC, as shown by Paul Cohen
Paul Cohen (mathematician)
Paul Joseph Cohen was an American mathematician best known for his proof of the independence of the continuum hypothesis and the axiom of choice from Zermelo–Fraenkel set theory, the most widely accepted axiomatization of set theory.-Early years:Cohen was born in Long Branch, New Jersey, into a...
and Kurt Gödel
Kurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
.
Martin's axiom
Martin's axiom
In 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...
together with the negation of 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...
is undecidable in ZFC.
Assuming that ZFC is consistent, the existence of large cardinal numbers, such as inaccessible cardinal
Inaccessible cardinal
In set theory, an uncountable regular cardinal number is called weakly inaccessible if it is a weak limit cardinal, and strongly inaccessible, or just inaccessible, if it is a strong limit cardinal. Some authors do not require weakly and strongly inaccessible cardinals to be uncountable...
s, Mahlo cardinal
Mahlo cardinal
In mathematics, a Mahlo cardinal is a certain kind of large cardinal number. Mahlo cardinals were first described by . As with all large cardinals, none of these varieties of Mahlo cardinals can be proved to exist by ZFC ....
s etc., cannot be proved in ZFC. On the other hand, few working set theorists expect their existence to be disproved.
Set theory of the real lineSet theory of the real lineSet theory of the real line is an area of mathematics concerned with the application of set theory to aspects of the real numbers.For example, one knows that all countable sets of reals are null, i.e. have Lebesgue measure 0; one might therefore ask the least possible size of a setwhich is not...
There are many cardinal invariants of the real line, connected with measure theory and statements related to the Baire category theoremBaire category theorem
The Baire category theorem is an important tool in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space....
whose exact values are independent of ZFC (in a stronger sense than that the continuum hypothesis is in ZFC. While nontrivial relations can be proved between them, most cardinal invariants can be any regular cardinal
Regular cardinal
In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. So, crudely speaking, a regular cardinal is one which cannot be broken into a smaller collection of smaller parts....
between
and ). This is a major area of study in set theoretic real analysis. Martin's axiom has a tendency to set most interesting cardinal invariants equal to .Order theoryOrder theoryOrder theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...
The answer to Suslin's problemSuslin's problem
In mathematics, Suslin's problem is a question about totally ordered sets posed by Mikhail Yakovlevich Suslin in a work published posthumously in 1920....
is independent of ZFC. The diamond principle ◊ proves the existence of a Suslin line, while Martin's axiom
Martin's axiom
In 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...
+ the negation of the continuum hypothesis proves that no Suslin line exists.
Existence of Kurepa trees is independent of ZFC.
Group theoryGroup theoryIn 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...
In 1973, Saharon ShelahSaharon 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:...
showed that the Whitehead problem
Whitehead problem
In group theory, a branch of abstract algebra, the Whitehead problem is the following question:Shelah proved that Whitehead's problem is undecidable within standard ZFC set theory.-Refinement:...
("is every abelian group
Abelian group
In 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 Ext1(A, Z) = 0 a free abelian group?") is independent of ZFC. A group with Ext1(A, Z) = 0 which is not free abelian is called a Whitehead group; Martin's Axiom
Martin's axiom
In 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...
+ the negation of the continuum hypothesis proves the existence of a Whitehead group, while the axiom of constructibility
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 constructible universe, respectively.- Implications :The axiom of...
proves that no Whitehead group exists.
Measure theory
A stronger version of Fubini's theoremFubini's theorem
In mathematical analysis Fubini's theorem, named after Guido Fubini, is a result which gives conditions under which it is possible to compute a double integral using iterated integrals. As a consequence it allows the order of integration to be changed in iterated integrals.-Theorem...
for positive functions, where the function is no longer assumed to be measurable but merely that the two iterated integrals are well defined and exist, is independent of ZFC. On the one hand, 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...
implies that there exists a function on the unit square whose iterated integrals are not equal — the function is simply the indicator function of an ordering of [0, 1] equivalent to a well ordering of the cardinal
. A similar example can be constructed using Martin's axiomMartin's axiom
In 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...
. On the other hand, the consistency of the strong Fubini theorem was first shown by Friedman
Harvey Friedman
Harvey Friedman is a mathematical logician at Ohio State University in Columbus, Ohio. He is noted especially for his work on reverse mathematics, a project intended to derive the axioms of mathematics from the theorems considered to be necessary...
. It can also be deduced from a variant of Freiling's axiom of symmetry
Freiling's axiom of symmetry
Freiling's axiom of symmetry is a set-theoretic axiom proposed by Chris Freiling. It is based on intuition of Stuart Davidsonbut the mathematics behind it goes back to Wacław Sierpiński....
.
Functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
Garth Dales and Robert M. SolovayRobert M. Solovay
Robert Martin Solovay is an American mathematician specializing in set theory.Solovay earned his Ph.D. from the University of Chicago in 1964 under the direction of Saunders Mac Lane, with a dissertation on A Functorial Form of the Differentiable Riemann–Roch theorem...
proved in 1976 that Kaplansky's conjecture, namely that there exists a discontinuous 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...
from the Banach algebra
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space...
C(X) (where X is some infinite compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
Hausdorff space
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
) into any other Banach algebra, was independent of ZFC. Assuming 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...
enables proving that for any infinite X there exists such a homomorphism into any Banach algebra.
Charles Akemann and Nik Weaver showed in 2003 that the statement "there exists a counterexample 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....
which is generated by
, elements" is independent of ZFC.Miroslav Bačák and Petr Hájek
Petr Hájek
Petr Hájek is a Czech scientist in the area of mathematical logic and a professor of mathematics. He works at the Institute of Computer Science at the Academy of Sciences of the Czech Republic and worked as a lecturer at the Faculty of Mathematics and Physics at the Charles University in Prague...
proved in 2008 that the statement "every Asplund space of density character
has a renorming with the Mazur intersection property" is independent of ZFC. The result is shown using Martin's maximumMartin's maximum
In set theory, Martin's maximum, introduced by , is a generalization of the proper forcing axiom, which is in turn a generalization of Martin's axiom....
axiom, while Mar Jiménez and José Pedro Moreno (1997) had presented a counterexample under 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...
.
As shown by Ilijas Farah and N. Christopher Phillips, the existence of outer automorphisms of the Calkin algebra
Calkin algebra
In functional analysis, the Calkin algebra, named after John Wilson Calkin, is the quotient of B, the ring of bounded linear operators on a separable infinite-dimensional Hilbert space H, by the ideal K of compact operators....
depends on set theoretic assumptions beyond ZFC.