List of statements undecidable in ZFC

The following is a list of mathematical statements that are undecidable in ZFC (the Zermelo–Fraenkel axioms plus the axiom of choice), assuming that ZFC is consistent.

Functional analysis

Charles Akemann and Nik Weaver showed in 2003 that the statement "there exists a counterexample to Naimark's problem which is generated by $aleph\_1$, elements" is independent of ZFC.

Garth Dales and Robert M. Solovay proved in 1976 that Kaplansky's conjecture as to whether there exists a discontinuous homomorphism from the Banach algebra "C(X)" (where "X" is some infinite compact Hausdorff topological space) into any other Banach algebra was independent of ZFC, but that the continuum hypothesis proves that for any infinite "X" there exists such a homomorphism into any Banach algebra.

Measure theory

A stronger version of Fubini's 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 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 $omega\_1$. A similar example can be constructed using Martin's axiom. On the other hand, the consistency of the strong Fubini theorem was first showed by Friedman [Harvey Freidman, "A Consistent Fubini-Tonelli Theorem for Nonmeasureable Functions", Illinois J. Math., Vol. 24 (1980), no. 3, 390–395] . It can also be deduced from a variant of Freiling's axiom of symmetry [Chris Freiling, "Axioms of symmetry: throwing darts at the real number line", J. Symbolic Logic 51 (1986), no. 1, 190–200.] .

Axiomatic set theory

The consistency of ZFC was the first statement shown to be undecidable in ZFC.

The axiom "V=L" (that all sets are constructible) implies the generalized continuum hypothesis (which states that ℵ_"n" = ℶ_n 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.

The existence of large cardinal numbers, such as inaccessible cardinals, Mahlo cardinals etc., cannot be proved in ZFC, and few working set theorists expect them to be disproved. However it is not possible to formalize in ZFC a proof that ZFC cannot refute the existence of large cardinals (even under the added hypothesis that ZFC is itself consistent).

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 non-trivial relations can be proved between them, most cardinal invariants can be any regular cardinal between $aleph\_1$ and $2^\{aleph\_0\}$). 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 $2^\{aleph\_0\}$.

Group theory

The Whitehead problem ("is every abelian group "A" with Ext¹(A, Z) = 0 a free abelian group?") is independent of ZFC, as shown in 1973 by Saharon Shelah. 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 V=L proves that no Whitehead group exists.

Order theory

The answer to Suslin's problem is independent of ZFC. ◊ proves the existence of a Suslin line, while Martin's axiom + the negation of the continuum hypothesis proves that no Suslin line exists.

References