List of statements undecidable in ZFC

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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.

Axiomatic set theory[edit]

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 undecidable in ZFC (shown by Paul Cohen and Kurt Gödel, and, in the case of MA + ¬CH, Solovay and Tennenbaum):

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.

Set theory of the real line[edit]

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 20). 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 20.

Whether all \aleph_1-dense subsets of the real line are isomorphic is undecidable in ZFC.[2]

Borel's conjecture, which states that every strong measure zero set is countable, is undecidable.

Order theory[edit]

The answer to Suslin's problem is undecidable in ZFC.[3] The diamond principle ◊ proves the existence of a Suslin line, while MA + ¬CH implies EATS (every Aronszajn tree is special),[4] which in turn impliss (but is not equivalent to)[5] the nonexistence of Suslin lines. Ronald Jensen proved that CH does not imply the existence of a Suslin line.[6]

Existence of Kurepa trees is undecidable in ZFC, assuming consistency of an inaccessible cardinal.[7]

Existence of a partition of \omega_2 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.[8][9][10]

Group theory[edit]

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.[11] 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.[12][13]

Measure theory[edit]

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.[14] It can also be deduced from a variant of Freiling's axiom of symmetry.[15]

Topology[edit]

Normal Moore space conjecture

S- and L- spaces

Functional analysis[edit]

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[16] and N. Christopher Phillips and Nik Weaver,[17] the existence of outer automorphisms of the Calkin algebra depends on set theoretic assumptions beyond ZFC.

References[edit]

  1. ^ Kunen, Kenneth (1980). Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9. 
  2. ^ Baumgartner, J., All \aleph_1-dense sets of reals can be isomorphic, Fund. Math. 79, pp.101 -- 106, 1973
  3. ^ 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. 
  4. ^ Baumgartner, J., J. Malitz, and W. Reiehart, Embedding trees in the rationals, Proc. Nat. Acad. Science, U.S.A., 67, pp. 1746 -- 1753, 1970
  5. ^ Shelah, S., Free limits of forcing and more on Aronszajn trees, Israel Journal of Mathematics, 40, pp. 1 -- 32, 1971
  6. ^ Devlin, K., and H. Johnsbraten, The Souslin Problem, Lecture Notes on Mathematics 405, Springer, 1974
  7. ^ Silver, J., The independence of Kurepa's conjecture and two-cardinal conjectures in model theory, in Axiomatic Set Theory, Proc. Symp, in Pure Mathematics (13) pp. 383 - 390, 1967
  8. ^ Shelah, S., Proper and Improper Forcing, Springer 1992
  9. ^ Schlindwein, Chaz, Shelah's work on non-semiproper iterations I, Archive for Mathematical Logic (47) 2008 pp. 579 -- 606
  10. ^ Schlindwein, Chaz, Shelah's work on non-semiproper iterations II, Journal of Symbolic Logic (66) 2001, pp. 1865 -- 1883
  11. ^ Shelah, S. (1974). "Infinite Abelian groups, Whitehead problem and some constructions". Israel Journal of Mathematics 18: 243–256. doi:10.1007/BF02757281. MR 0357114. 
  12. ^ Shelah, S., Whitehead groups may not be free even assuming CH I, Israel Journal of Mathematics (28) 1972
  13. ^ Shelah, S., Whitehead groups may not be free even assuming CH II, Israel Journal of Mathematics (350 1980
  14. ^ Friedman, Harvey (1980). "A Consistent Fubini-Tonelli Theorem for Nonmeasurable Functions". Illinois J. Math. 24 (3): 390–395. MR 573474. 
  15. ^ 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. 
  16. ^ Farah, Ilijas (2007). "All automorphisms of the Calkin algebra are inner". arXiv:0705.3085.
  17. ^ 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.