List of statements independent of ZFC

The

. 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,

.

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

• the
  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

) 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
  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

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

Order theory

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

Existence of a partition of the ordinal number ${\displaystyle \omega _{2}}$ 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,

+ ¬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

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

Topology

The Normal Moore Space conjecture, namely that every normal Moore space is metrizable, can be disproven assuming CH or MA + ¬CH, 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.^{[citation needed]}

Various assertions about ${\displaystyle P(\omega )/}$finite, P-points, Q-points, ...^{[further explanation needed]}

The existence of an

Functional analysis

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.^[21]

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^[22] and N. Christopher Phillips and Nik Weaver,^[23] the existence of outer automorphisms of the Calkin algebra depends on set theoretic assumptions beyond ZFC.

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.^[25]

References

External links

• What are some reasonable-sounding statements that are independent of ZFC?, mathoverflow.net