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 ℵ₁ elements" is independent of ZFC.

Axiomatic set theory

The continuum hypothesis (which states that ℵ₁ = ℶ₁), and the generalized continuum hypothesis (which states that ℵ_n = ℶ_n for every n) are independent of ZFC (as shown by Paul Cohen and Kurt Gödel), as is the combinatorial statement ◊ (which implies CH).

The existence of large cardinal numbers, such as inaccessible cardinals, Mahlo cardinals etc., can neither be proven nor disproven in ZFC.

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.