List of statements undecidable in ZFC
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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.
Garth Dales and Robert Solovay proved in 1976 that Kaplanksy'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
The existence of strong versions of Fubini's Theorem, 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. Martin's Axiom implies that there exists a function on the unit square whose iterated integrals are not equal, while as a variant of Freiling's Axiom of Symmetry implies that in fact a strong Fubini type theorem for [0, 1] does hold, and whenever the two iterated integrals exist they are equal.
Axiomatic set theory
The continuum hypothesis (which states that ℵ1 = ℶ1), and the generalized continuum hypothesis (which states that ℵn = ℶn 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.
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 <math>\aleph_1<math> and <math>2^{\aleph_0}<math>). 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 <math>2^{\aleph_0}<math>.
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.
Other
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.