List of Statements Independent of ZFC

Themathematical statements discussed below are provablyindependent ofZFC (the Zermelo–Fraenkel axioms plus theaxiom of choice, the canonicalaxiomatic set theory of contemporary mathematics), assuming that ZFC isconsistent. A statement is independent of ZFC (sometimes phrased "undecidable in ZFC") if it can neither be proven nor disproven from the axioms of ZFC.

Axiomatic set theory [ edit ]

In 1931,Kurt Gödel proved the first ZFC independence result, namely that the consistency of ZFC itself was independent of ZFC ( Gödel's second incompleteness theorem ).

The following statements are independent of ZFC, among others:

• the consistency of ZFC;
• thecontinuum hypothesis or CH (Gödel produced a model of ZFC in which CH is true, showing that CH cannot be disproven in ZFC;Paul Cohen later invented the method offorcing to exhibit a model of ZFC in which CH fails, showing that CH cannot be proven in ZFC. The following four independence results are also due to Gödel/Cohen.);
• the generalized continuum hypothesis (GCH);
• a related independent statement is that if a set x has fewer elements than y , then x also has fewersubsets than y . In particular, this statement fails when the cardinalities of the power sets of x and y coincide;
• the axiom of constructibility ( V = L );
• thediamond principle (◊);
• Martin's axiom (MA);
• MA + ¬ CH (independence shown bySolovay andTennenbaum).

We have the following chains of implications:

V = L → ◊ → CH, V = L → GCH → CH, CH → MA,

and (see section on order theory):

◊ → ¬SH, MA + ¬ CH →EATS → SH.

Several statements related to the existence oflarge cardinals cannot be proven in ZFC (assuming ZFC is consistent). These are independent of ZFC provided that they are consistent with ZFC, which most working set theorists believe to be the case. These statements are strong enough to imply the consistency of ZFC. This has the consequence (via Gödel's second incompleteness theorem ) that their consistency with ZFC cannot be proven in ZFC (assuming ZFC is consistent). The following statements belong to this class:

• Existence of inaccessible cardinals
• Existence ofMahlo cardinals
• Existence ofmeasurable cardinals (first conjectured byUlam)
• Existence of supercompact cardinals

The following statements can be proven to be independent of ZFC assuming the consistency of a suitable large cardinal:

• Proper forcing axiom
• Open coloring axiom
• Martin's maximum
• Existence of 0 #
• Singular cardinals hypothesis
• Projective determinacy (and even the fullaxiom of determinacy if theaxiom of choice is not assumed)

Set theory of the real line [ edit ]

There are manycardinal invariants of the real line, connected withmeasure theory and statements related to the Baire category theorem , whose exact values are independent of ZFC. While nontrivial relations can be proved between them, most cardinal invariants can be anyregular cardinal between ℵ 1 and 2 ℵ 0 . This is a major area of study in the set theory of the real line (seeCichon diagram). MA has a tendency to set most interesting cardinal invariants equal to 2 ℵ 0 .

A subset X of the real line is a strong measure zero set if to every sequence (ε n ) of positive reals there exists a sequence of intervals ( I n ) which covers X and such that I n has length at most ε n . Borel's conjecture, that every strong measure zero set is countable, is independent of ZFC.

A subset X of the real line is -dense if every open interval contains -many elements of X . Whether all -dense sets are order-isomorphic is independent of ZFC.

Order theory [ edit ]

Suslin's problem asks whether a specific short list of properties characterizes the ordered set of real numbers R . This is undecidable in ZFC.A Suslin line is an ordered set which satisfies this specific list of properties but is not order-isomorphic to R . Thediamond principle ◊ proves the existence of a Suslin line, while MA + ¬CH implies EATS ( every Aronszajn tree is special ),which in turn implies (but is not equivalent to)the nonexistence of Suslin lines.Ronald Jensen proved that CH does not imply the existence of a Suslin line.

Existence ofKurepa trees is independent of ZFC, assuming consistency of an inaccessible cardinal .

Existence of a partition of theordinal number into two colors with no monochromatic uncountable sequentially closed subset is independent of ZFC, ZFC + CH, and ZFC + ¬CH, assuming consistency of aMahlo cardinal.This theorem ofShelah answers a question ofH. Friedman.

Abstract algebra [ edit ]

In 1973,Saharon Shelah showed that theWhitehead problem ("is everyabelian group A withExt 1 (A, Z ) = 0 afree abelian group?") is independent of ZFC.An abelian group with Ext 1 (A, Z ) = 0 is called a Whitehead group;MA + ¬CH proves the existence of a non-free Whitehead group, while V = L proves that all Whitehead groups are free. In one of the earliest applications of properforcing, Shelah constructed a model of ZFC + CH in which there is a non-free Whitehead group.

Consider the ring A = R [ x , y , z ] of polynomials in three variables over the real numbers and itsfield of fractions M = R ( x , y , z ). Theprojective dimension of M as A -module is either 2 or 3, but it is independent of ZFC whether it is equal to 2; it is equal to 2 if and only if CH holds.

Adirect product of countably manyfields hasglobal dimension 2 if and only if the continuum hypothesis holds.

Number theory [ edit ]

One can write down a concrete polynomial p ∈ Z [ x 1 ,... x 9 ] such that the statement "there are integers m 1 ,..., m 9 with p ( m 1 ,..., m 9 )=0" can neither be proven nor disproven in ZFC (assuming ZFC is consistent).

[ circular reference ]

This follows fromYuri Matiyasevich's resolution of Hilbert's tenth problem ; the polynomial is constructed so that it has an integer root if and only if ZFC is inconsistent.

Measure theory [ edit ]

A stronger version ofFubini's theorem for positive functions, where the function is no longer assumed to bemeasurable but merely that the two iterated integrals are well defined and exist, is independent of ZFC. On the one hand, CH implies that there exists a function on the unit square whose iterated integrals are not equal — the function is simply theindicator function of an ordering of [0, 1] equivalent to awell ordering of the cardinal ω 1 . A similar example can be constructed usingMA. On the other hand, the consistency of the strong Fubini theorem was first shown byFriedman.It can also be deduced from a variant of Freiling's axiom of symmetry .

Topology [ edit ]

The Normal Moore Space conjecture, namely that everynormal Moore space ismetrizable, can be disproven assuming CH or MA + ¬CH, and can be proven assuming a certain axiom which implies the existence of large cardinals. Thus, granted large cardinals, the Normal Moore Space conjecture is independent of ZFC.

Various assertions about finite, P-points, Q-points,...

S- and L- spaces

Functional analysis [ edit ]

Garth Dales andRobert M. Solovay proved in 1976 that Kaplansky's conjecture , namely that everyalgebra homomorphism from theBanach algebra C(X) (where X is somecompact Hausdorff space) into any other Banach algebra must be continuous, is independent of ZFC. CH implies that for any infinite X there exists a discontinuous homomorphism into any Banach algebra.

Consider the algebra B ( H ) of bounded linear operators on the infinite-dimensionalseparable Hilbert space H . Thecompact operators form a two-sided ideal in B ( H ). The question of whether this ideal is the sum of two properly smaller ideals is independent of ZFC, as was proved byAndreas Blass andSaharon Shelah in 1987.

Charles Akemann andNik Weaver showed in 2003 that the statement "there exists a counterexample toNaimark's problem which is generated by ℵ 1 , elements" is independent of ZFC.

Miroslav Bačák andPetr Hájek proved in 2008 that the statement "everyAsplund space of density character ω 1 has a renorming with the Mazur intersection property " is independent of ZFC. The result is shown usingMartin's maximum axiom, while Mar Jiménez and José Pedro Moreno (1997) had presented a counterexample assuming CH.

As shown byIlijas Farah and N. Christopher Phillips andNik Weaver,the existence of outer automorphisms of theCalkin algebra depends on set theoretic assumptions beyond ZFC.

Model theory [ edit ]

Chang's conjecture is independent of ZFC assuming the consistency of anErdős cardinal.

Computability theory [ edit ]

Marcia Groszek andTheodore Slaman gave examples of statements independent of ZFC concerning the structure of the Turing degrees. In particular, whether there exists a maximally independent set of degrees of size less than continuum.

References [ edit ]