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.
- The axiom of constructibility (V = L);
- The generalized continuum hypothesis (GCH);
- The continuum hypothesis (CH);
- The diamond principle (◊);
- Martin's axiom (MA);
- MA + ¬CH.
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 can be disproved.
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. 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 (see Cichon diagram). MA has a tendency to set most interesting cardinal invariants equal to 2ℵ0.
Whether all -dense subsets of the real line are isomorphic is undecidable in ZFC.
Borel's conjecture, which states that every strong measure zero set is countable, is undecidable.
The answer to Suslin's problem is undecidable in ZFC. The diamond principle ◊ proves the existence of a Suslin line, while MA + ¬CH implies EATS (every Aronszajn tree is special), which in turn impliss (but is not equivalent to) the nonexistence of Suslin lines. Ronald Jensen proved that CH does not imply the existence of a Suslin line.
Existence of a partition of into two colors with no monochromatic uncountable sequentially closed subset is undecidable in ZFC, ZFC + CH, and ZFC + ¬CH, assuming consistency of a Mahlo cardinal. This theorem of Shelah answers a question of H. Friedman.
In 1973, Saharon Shelah showed that the Whitehead problem ("is every abelian group A with Ext1(A, Z) = 0 a free abelian group?") is undecidable in ZFC. An abelian group with Ext1(A, Z) = 0 is called a Whitehead group; MA + ¬CH proves the existence of a non-free Whitehead group, while V = L proves that all Whitehead groups are free. 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.
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 undecidable in 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. It can also be deduced from a variant of Freiling's axiom of symmetry.
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Normal Moore space conjecture
S- and L- spaces
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.
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.
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- Solovay, R. M.; Tennenbaum, S. (1971). "Iterated Cohen extensions and Souslin's problem". Ann. Of Math. (2) (Annals of Mathematics) 94 (2): 201–245. doi:10.2307/1970860. JSTOR 1970860.
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- Devlin, K., and H. Johnsbraten, The Souslin Problem, Lecture Notes on Mathematics 405, Springer, 1974
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- Schlindwein, Chaz, Shelah's work on non-semiproper iterations I, Archive for Mathematical Logic (47) 2008 pp. 579 -- 606
- Schlindwein, Chaz, Shelah's work on non-semiproper iterations II, Journal of Symbolic Logic (66) 2001, pp. 1865 -- 1883
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- Shelah, S., Whitehead groups may not be free even assuming CH I, Israel Journal of Mathematics (28) 1972
- Shelah, S., Whitehead groups may not be free even assuming CH II, Israel Journal of Mathematics (350 1980
- Friedman, Harvey (1980). "A Consistent Fubini-Tonelli Theorem for Nonmeasurable Functions". Illinois J. Math. 24 (3): 390–395. MR 573474.
- Freiling, Chris (1986). "Axioms of symmetry: throwing darts at the real number line". Journal of Symbolic Logic 51 (1): 190–200. JSTOR 2273955. MR 830085.
- Farah, Ilijas (2007). "All automorphisms of the Calkin algebra are inner". arXiv:0705.3085.
- Phillips, N. C.; Weaver, N. (2007). "The Calkin algebra has outer automorphisms". Duke Mathematical Journal 139 (1): 185–202. doi:10.1215/S0012-7094-07-13915-2.