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Axiomatic set theory In 1931, Kurt Gödel proved the first ZFC undecidability result, namely that the consistency of ZFC itself was undecidable in ZFC. Moreover the following statements are independent of ZFC (shown by Paul Cohen and Kurt Gödel): The axiom of constructibility (V = L); The generalized continuum hypothesis (GCH); The continuum hypothesis (CH); The diamond principle (◊); Martin's axiom (MA); MA + ¬CH.[1] Note that we have the following chains of implication: V = L → ◊ V = L → GCH → CH. Assuming that ZFC is consistent, the existence of large cardinal numbers, such as inaccessible cardinals, Mahlo cardinals etc., cannot be proven 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 CH is in ZFC. While nontrivial relations can be proved between them, most cardinal invariants can be any regular cardinal between ℵ1 and 2ℵ0). This is a major area of study in set theoretic real analysis. MA has a tendency to set most interesting cardinal invariants equal to 2ℵ0. Order theory The answer to Suslin's problem is independent of ZFC.[2] The diamond principle ◊ proves the existence of a Suslin line, while MA + ¬CH 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.[3] A group with Ext1(A, Z) = 0 which is not free abelian is called a Whitehead group; MA + ¬CH proves the existence of a Whitehead group, while V = L 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, CH 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 MA. On the other hand, the consistency of the strong Fubini theorem was first shown by Friedman.[4] It can also be deduced from a variant of Freiling's axiom of symmetry.[5] 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. CH implies 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 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[6] and N. Christopher Phillips and Nik Weaver,[7] the existence of outer automorphisms of the Calkin algebra depends on set theoretic assumptions beyond ZFC. |
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