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 statement is independent of ZFC (sometimes phrased "undecidable in ZFC") if it can neither be proven nor disproven from the axioms of ZFC.

Axiomatic set theory

In 1931, Kurt Gödel proved his incompleteness theorems, establishing that many mathematical theories, including ZFC, cannot prove their own consistency. Assuming ω-consistency of such a theory, the consistency statement can also not be disproven, meaning it is independent. A few years later, other arithmetic statements were defined that are independent of any such theory, see for example Rosser's trick.

The following set theoretic statements are independent of ZFC, among others:

the continuum hypothesis or CH (Gödel produced a model of ZFC in which CH is true, showing that CH cannot be disproven in ZFC; Paul Cohen later invented the method of forcing to exhibit a model of ZFC in which CH fails, showing that CH cannot be proven in ZFC. The following four independence results are also due to Gödel/Cohen.);
the generalized continuum hypothesis (GCH);
a related independent statement is that if a set x has fewer elements than y, then x also has fewer subsets than y. In particular, this statement fails when the cardinalities of the power sets of x and y coincide;
the axiom of constructibility (V = L);
the diamond principle (◊);
Martin's axiom (MA);
MA + ¬CH (independence shown by Solovay and Tennenbaum).^[1]
Every Aronszajn tree is special (EATS);

We have the following chains of implications:

V = L → ◊ → CH,

V = L → GCH → CH,

CH → MA,

and (see section on order theory):

◊ → ¬SH,

MA + ¬CH → EATS → SH.

Several statements related to the existence of large cardinals cannot be proven in ZFC (assuming ZFC is consistent). These are independent of ZFC provided that they are consistent with ZFC, which most working set theorists believe to be the case. These statements are strong enough to imply the consistency of ZFC. This has the consequence (via Gödel's second incompleteness theorem) that their consistency with ZFC cannot be proven in ZFC (assuming ZFC is consistent). The following statements belong to this class:

Existence of inaccessible cardinals
Existence of Mahlo cardinals
Existence of measurable cardinals (first conjectured by Ulam)
Existence of supercompact cardinals

The following statements can be proven to be independent of ZFC assuming the consistency of a suitable large cardinal:

Proper forcing axiom
Open coloring axiom
Martin's maximum
Existence of 0^#
Singular cardinals hypothesis
Projective determinacy (and even the full axiom of determinacy if the axiom of choice is not assumed)

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. While nontrivial relations can be proved between them, most cardinal invariants can be any regular cardinal between ℵ₁ and 2^ℵ₀. This is a major area of study in the set theory of the real line (see Cichon diagram). MA has a tendency to set most interesting cardinal invariants equal to 2^ℵ₀.

A subset X of the real line is a strong measure zero set if to every sequence (ε_n) of positive reals there exists a sequence of intervals (I_n) which covers X and such that I_n has length at most ε_n. Borel's conjecture, that every strong measure zero set is countable, is independent of ZFC.

A subset X of the real line is $\aleph _{1}$ -dense if every open interval contains $\aleph _{1}$ -many elements of X. Whether all $\aleph _{1}$ -dense sets are order-isomorphic is independent of ZFC.^[2]

Order theory

Suslin's problem asks whether a specific short list of properties characterizes the ordered set of real numbers R. This is undecidable in ZFC.^[3] A Suslin line is an ordered set which satisfies this specific list of properties but is not order-isomorphic to R. The diamond principle ◊ proves the existence of a Suslin line, while MA + ¬CH implies EATS (every Aronszajn tree is special),^[4] which in turn implies (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 independent of ZFC, assuming consistency of an inaccessible cardinal.^[7]

Existence of a partition of the ordinal number $\omega _{2}$ into two colors with no monochromatic uncountable sequentially closed subset is independent of ZFC, ZFC + CH, and ZFC + ¬CH, assuming consistency of a Mahlo cardinal.^[8]^[9]^[10] This theorem of Shelah answers a question of H. Friedman.

Abstract algebra

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.^[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]

Consider the ring A = R[x,y,z] of polynomials in three variables over the real numbers and its field of fractions M = R(x,y,z). The projective dimension of M as A-module is either 2 or 3, but it is independent of ZFC whether it is equal to 2; it is equal to 2 if and only if CH holds.^[14]

A direct product of countably many fields has global dimension 2 if and only if the continuum hypothesis holds.^[15]

Number theory

One can write down a concrete polynomial p ∈ Z[x₁, ..., x₉] such that the statement "there are integers m₁, ..., m₉ with p(m₁, ..., m₉) = 0" can neither be proven nor disproven in ZFC (assuming ZFC is consistent). 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.^[16]

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.^[17] It can also be deduced from a variant of Freiling's axiom of symmetry.^[18]

Topology

The Normal Moore Space conjecture, namely that every normal Moore space is metrizable, can be disproven assuming the continuum hypothesis or assuming both Martin's axiom and the negation of the continuum hypothesis, and can be proven assuming a certain axiom which implies the existence of large cardinals. Thus, granted large cardinals, the Normal Moore Space conjecture is independent of ZFC.^[19]

The existence of an S-space is independent of ZFC. In particular, it is implied by the existence of a Suslin line.^[20]

Functional analysis

Garth Dales and Robert M. Solovay proved in 1976 that Kaplansky's conjecture, namely that every algebra homomorphism from the Banach algebra C(X) (where X is some compact Hausdorff space) into any other Banach algebra must be continuous, is independent of ZFC. CH implies that for any infinite X there exists a discontinuous homomorphism into any Banach algebra.^[21]

Consider the algebra B(H) of bounded linear operators on the infinite-dimensional separable Hilbert space H. The compact operators form a two-sided ideal in B(H). The question of whether this ideal is the sum of two properly smaller ideals is independent of ZFC, as was proved by Andreas Blass and Saharon Shelah in 1987.^[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.

Wetzel's problem, which asks if every set of analytic functions which takes at most countably many distinct values at every point is necessarily countable, is true if and only if the continuum hypothesis is false.^[25]

Model theory

Chang's conjecture is independent of ZFC assuming the consistency of an Erdős cardinal.

Computability theory

Marcia Groszek and Theodore Slaman gave examples of statements independent of ZFC concerning the structure of the Turing degrees. In particular, whether there exists a maximally independent set of degrees of size less than continuum.^[26]

References

^ Kunen, Kenneth (1980). Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.
^ Baumgartner, J., All $\aleph _{1}$ -dense sets of reals can be isomorphic, Fund. Math. 79, pp.101 – 106, 1973
^ Solovay, R. M.; Tennenbaum, S. (1971). "Iterated Cohen extensions and Souslin's problem". Annals of Mathematics. Second Series. 94 (2): 201–245. doi:10.2307/1970860. JSTOR 1970860.
^ Baumgartner, J., J. Malitz, and W. Reiehart, Embedding trees in the rationals, Proc. Natl. Acad. Sci. U.S.A., 67, pp. 1746 – 1753, 1970
^ Shelah, S. (1981). "Free limits of forcing and more on Aronszajn trees". Israel Journal of Mathematics. 38 (4): 315–334. doi:10.1007/BF02762777.
^ Devlin, K., and H. Johnsbraten, The Souslin Problem, Lecture Notes on Mathematics 405, Springer, 1974
^ 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
^ Shelah, S., Proper and Improper Forcing, Springer 1992
^ 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
^ Shelah, S. (1974). "Infinite Abelian groups, Whitehead problem and some constructions". Israel Journal of Mathematics. 18 (3): 243–256. doi:10.1007/BF02757281. MR 0357114.
^ Shelah, S. (1972). "Whitehead groups may be not free, even assuming CH, I". Israel Journal of Mathematics. 28 (3): 193–204. doi:10.1007/BF02759809.
^ Shelah, S. (1980). "Whitehead groups may not be free even assuming CH, II". Israel Journal of Mathematics. 35 (4): 257–285. doi:10.1007/BF02760652.
^ Barbara L. Osofsky (1968). "Homological dimension and the continuum hypothesis" (PDF). Transactions of the American Mathematical Society. 132: 217–230. doi:10.1090/s0002-9947-1968-0224606-4.
^ Barbara L. Osofsky (1973). Homological Dimensions of Modules. American Mathematical Soc. p. 60. ISBN 978-0-8218-1662-2.
^
See e.g.:
- James P. Jones (1980). "Undecidable diophantine equations". Bull. Amer. Math. Soc. 3 (2): 859–862. doi:10.1090/s0273-0979-1980-14832-6.
- Carl, M.; Moroz, B. (2014). "On a Diophantine Representation of the Predicate of Provability". Journal of Mathematical Sciences. 199 (199): 36–52. doi:10.1007/s10958-014-1830-2. hdl:21.11116/0000-0004-1E89-1. S2CID 34618563.
For a summary of the argument, see Hilbert's tenth problem § Applications.
^ Friedman, Harvey (1980). "A Consistent Fubini-Tonelli Theorem for Nonmeasurable Functions". Illinois J. Math. 24 (3): 390–395. doi:10.1215/ijm/1256047607. MR 0573474.
^ Freiling, Chris (1986). "Axioms of symmetry: throwing darts at the real number line". Journal of Symbolic Logic. 51 (1): 190–200. doi:10.2307/2273955. JSTOR 2273955. MR 0830085. S2CID 38174418.
^ Nyikos, Peter J. (2001). "A history of the normal Moore space problem". Handbook of the History of General Topology. Vol. 3. Dordrecht: Kluwer Academic Publishers. pp. 1179–1212. doi:10.1007/978-94-017-0470-0_7. ISBN 0-7923-6970-X. MR 1900271.
^ Todorcevic, Stevo (1989). Partition problems in topology. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-5091-6.
^ H. G. Dales; W. H. Woodin (1987). An introduction to independence for analysts.
^ Judith Roitman (1992). "The Uses of Set Theory". Mathematical Intelligencer. 14 (1).
^ Farah, Ilijas (2011). "All automorphisms of the Calkin algebra are inner". Annals of Mathematics. Second Series. 173 (2): 619–661. arXiv:0705.3085. doi:10.4007/annals.2011.173.2.1.
^ Phillips, N. C.; Weaver, N. (2007). "The Calkin algebra has outer automorphisms". Duke Mathematical Journal. 139 (1): 185–202. arXiv:math/0606594. doi:10.1215/S0012-7094-07-13915-2. S2CID 13873756.
^ Erdős, P. (1964). "An interpolation problem associated with the continuum hypothesis". The Michigan Mathematical Journal. 11: 9–10. doi:10.1307/mmj/1028999028. MR 0168482..
^ Groszek, Marcia J.; Slaman, T. (1983). "Independence results on the global structure of the Turing degrees". Transactions of the American Mathematical Society. 277 (2): 579. doi:10.2307/1999225. JSTOR 1999225.