Themathematical statements discussed below are provablyindependent ofZFC (the canonicalaxiomatic set theory of contemporary mathematics, consisting of theZermelo–Fraenkel axioms plus theaxiom of choice), 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.
In 1931,Kurt Gödel proved hisincompleteness theorems, establishing that many mathematical theories, including ZFC, cannot prove their own consistency. Assumingω-consistency of such a theory, the consistency statement can also not be disproven, meaning it is independent. A few years later, other arithmetic statements were defined that are independent of any such theory, see for exampleRosser's trick.
The following set theoretic statements are independent of ZFC, among others:
We have the following chains of implications:
and (see section on order theory):
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 (viaGödel'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:
The following statements can be proven to be independent of ZFC assuming the consistency of a suitable large cardinal:
There are manycardinal invariants of the real line, connected withmeasure theory and statements related to theBaire category theorem, whose exact values are independent of ZFC. While nontrivial relations can be proved between them, most cardinal invariants can be anyregular cardinal betweenℵ1 and2ℵ0. 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 2ℵ0.
A subsetX of the real line is astrong measure zero set if to every sequence (εn) of positive reals there exists a sequence of intervals (In) which coversX and such thatIn has length at mostεn. Borel's conjecture, that every strong measure zero set is countable, is independent of ZFC.
A subsetX of the real line is-dense if every open interval contains-many elements ofX. Whether all-dense sets are order-isomorphic is independent of ZFC.[2]
Suslin's problem asks whether a specific short list of properties characterizes the ordered set of real numbersR. This is undecidable in ZFC.[3] ASuslin line is an ordered set which satisfies this specific list of properties but is not order-isomorphic toR. Thediamond principle ◊ proves the existence of a Suslin line, while MA + ¬CH implies EATS (every Aronszajn tree is special),[4] which in turn implies (but is not equivalent to)[5] the nonexistence of Suslin lines.Ronald Jensen proved that CH does not imply the existence of a Suslin line.[6]
Existence ofKurepa trees is independent of ZFC, assuming consistency of aninaccessible cardinal.[7]
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.[8][9][10] This theorem ofShelah answers a question ofH. Friedman.
In 1973,Saharon Shelah showed that theWhitehead problem ("is everyabelian groupA withExt1(A,Z) = 0 afree abelian group?") is independent of ZFC.[11] An abelian group with Ext1(A,Z) = 0 is called a Whitehead group;MA + ¬CH proves the existence of a non-free Whitehead group, whileV =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.[12][13]
Consider the ringA =R[x,y,z] of polynomials in three variables over the real numbers and itsfield of fractionsM =R(x,y,z). Theprojective dimension ofM asA-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.[14]
Adirect product of countably manyfields hasglobal dimension 2 if and only if the continuum hypothesis holds.[15]
One can write down a concrete polynomialp ∈Z[x1, ...,x9] such that the statement "there are integersm1, ...,m9 withp(m1, ...,m9) = 0" can neither be proven nor disproven in ZFC (assuming ZFC is consistent). This follows fromYuri Matiyasevich's resolution ofHilbert's tenth problem; the polynomial is constructed so that it has an integer root if and only if ZFC is inconsistent.[16]
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 ω1. A similar example can be constructed usingMA. On the other hand, the consistency of the strong Fubini theorem was first shown byFriedman.[17] It can also be deduced from a variant ofFreiling's axiom of symmetry.[18]
The Normal Moore Space conjecture, namely that everynormalMoore space ismetrizable, can be disproven assuming thecontinuum hypothesis or assuming bothMartin's axiom and the negation of the continuum hypothesis, and can be proven assuming a certain axiom which implies the existence oflarge cardinals. Thus, granted large cardinals, the Normal Moore Space conjecture is independent of ZFC.[19]
The existence of anS-space is independent of ZFC. In particular, it is implied by the existence of a Suslin line.[20]
Garth Dales andRobert M. Solovay proved in 1976 thatKaplansky's conjecture, namely that everyalgebra homomorphism from theBanach algebraC(X) (whereX is somecompactHausdorff space) into any other Banach algebra must be continuous, is independent of ZFC. CH implies that for any infiniteX there exists a discontinuous homomorphism into any Banach algebra.[21]
Consider the algebraB(H) ofbounded linear operators on the infinite-dimensionalseparableHilbert spaceH. Thecompact operators form a two-sided ideal inB(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.[22]
Charles Akemann andNik Weaver showed in 2003 that the statement "there exists a counterexample toNaimark's problem which is generated by ℵ1, elements" is independent of ZFC.
Miroslav Bačák andPetr Hájek proved in 2008 that the statement "everyAsplund space of density character ω1 has a renorming with theMazur 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[23] andN. Christopher Phillips andNik Weaver,[24] the existence of outer automorphisms of theCalkin algebra depends on set theoretic assumptions beyond ZFC.
Wetzel's problem, which asks if every set ofanalytic functions which takes at most countably many distinct values at every point is necessarily countable, is true if and only if the continuum hypothesis is false.[25]
Chang's conjecture is independent of ZFC assuming the consistency of anErdős cardinal.
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.[26]
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