List of statements independent of ZFC

The mathematical statements discussed below are provably independent of ZFC (the canonical axiomatic set theory of contemporary mathematics, consisting of the Zermelo–Fraenkel axioms plus the axiom of choice), assuming that ZFC is consistent.

A few years later, other arithmetic statements were defined that are independent of any such theory, see for example Rosser's trick.

This is a major area of study in the set theory of the real line (see Cichon diagram).

[2] Suslin's problem asks whether a specific short list of properties characterizes the ordered set of real numbers R. This is undecidable in ZFC.

[6] Existence of Kurepa trees is independent of ZFC, assuming consistency of an inaccessible cardinal.

In one of the earliest applications of proper forcing, Shelah constructed a model of ZFC + CH in which there is a non-free Whitehead group.

[14] A direct product of countably many fields has global dimension 2 if and only if the continuum hypothesis holds.

This follows from Yuri 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.

Thus, granted large cardinals, the Normal Moore Space conjecture is independent of ZFC.

[22] 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.

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 assuming CH.

As shown by Ilijas Farah[23] and N. Christopher Phillips and Nik Weaver,[24] the existence of outer automorphisms of the Calkin algebra depends on set theoretic assumptions beyond ZFC.

Marcia Groszek and Theodore Slaman gave examples of statements independent of ZFC concerning the structure of the Turing degrees.