List of statements undecidable in ZFC

The mathematical statements discussed below are provably 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.

Axiomatic set theory

In 1931, Kurt Gödel 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 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.

Assuming that ZFC is consistent, the existence of large cardinal numbers, such as inaccessible cardinals, Mahlo cardinals 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 $\aleph_1$ 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 $2^{\aleph_0}$.

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.

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 ω₁. 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 $\aleph_1$, elements" is independent of ZFC.

Miroslav Bačák and Petr Hájek proved in 2008 that the statement "every Asplund space of density character ω₁ 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.