The following is a list of mathematical statements that are undecidable in ZFC (the Zermelo-Fraenkel axioms plus the axiom of choice), assuming that ZFC is consistent. Mathematics is the study of quantity, structure, space and change. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ... In mathematics, the axiom of choice is an axiom of set theory. ...

Functional analysis

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. Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ... 2003 is a common year starting on Wednesday of the Gregorian calendar. ... In mathematics, Naimarks problem is a question in functional analysis. ... In the branch of mathematics known as set theory, aleph usually refers to a series of numbers used to represent the cardinality (or size) of infinite sets. ...

Garth Dales and Robert Solovay proved in 1976 that Kaplanksy's conjecture as to whether there exists a discontinuous homomorphism from the Banach algebra C(X) (where X is some infinite compact hausdorff topological space) into any other Banach algebra was independent of ZFC, but that the continuum hypothesis proves that for any infinite X there exists such a homomorphism into any Banach algebra. In functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. ... Several specialized usages of the terms compact and compactness exist. ... Felix Hausdorff (November 8, 1868 - January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory and functional analysis. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

Measure Theory

The existence of strong versions of Fubini's Theorem, 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. Martin's Axiom implies that there exists a function on the unit square whose iterated integrals are not equal, while as a variant of Freiling's Axiom of Symmetry implies that in fact a strong Fubini type theorem for [0, 1] does hold, and whenever the two iterated integrals exist they are equal. In mathematics, a measure is a function that assigns a number, e. ... In axiomatic set theory, Martins axiom is a statement which is independent of the usual axioms of ZFC Set Theory. ... Freilings axiom of symmetry (AX) is a set-theoretic axiom proposed by Chris Freiling. ...

Axiomatic set theory

The continuum hypothesis (which states that ℵ₁ = ℶ₁), and the generalized continuum hypothesis (which states that ℵ_n = ℶ_n for every n) are independent of ZFC (as shown by Paul Cohen and Kurt Gödel), as is the combinatorial statement ◊ (which implies CH). Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... In the branch of mathematics known as set theory, aleph usually refers to a series of numbers used to represent the cardinality (or size) of infinite sets. ... In mathematics, the Hebrew letter (aleph) with various subscripts represents various infinite cardinal numbers (see aleph number). ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... Paul Joseph Cohen (born April 2, 1934) is an American mathematician. ... Kurt Gödel Kurt Gödel [kurt gøːdl], (April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher of mathematics. ... In mathematics, and particularly in axiomatic set theory, ◊S (diamondsuit or diamond) is a certain family of combinatorial principles. ...

The existence of large cardinal numbers, such as inaccessible cardinals, Mahlo cardinals etc., can neither be proven nor disproven in ZFC. In mathematics, a cardinal is called a large cardinal if it belongs to a class of cardinals, the existence of which provably cannot be proved within the standard axiomatic set theory ZFC, if one assumes ZFC itself is consistent. ... In mathematics, a cardinal number k > (aleph-null) is called weakly inaccessible, or just inaccessible, if the following two conditions hold. ... In mathematics, a Mahlo cardinal is a certain kind of large cardinal number. ...

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 the continuum hypothesis is in ZFC. While non-trivial relations can be proved between them, most cardinal invariants can be any regular cardinal between and ). This is a major area of study in Set theoretic real analysis. Martin's axiom has a tendency to set most interesting cardinal invariants equal to . In mathematics, the Baire category theorem is an important tool in the study of complete spaces, such as Banach spaces and Hilbert spaces, that arise in topology and functional analysis. ... An infinite cardinal number κ is called regular if cf(κ) = κ, where cf is the cofinality operation. ...

Group theory

The Whitehead problem ("is every abelian group A with Ext¹(A, Z) = 0 a free abelian group?") is independent of ZFC, as shown in 1973 by Saharon Shelah. A group with Ext¹(A, Z) = 0 which is not free abelian is called a Whitehead group; Martin's Axiom + the negation of the continuum hypothesis proves the existence of a Whitehead group, while V=L proves that no Whitehead group exists. Group theory is that branch of mathematics concerned with the study of groups. ... In group theory, the Whitehead problem is the following question: Is every abelian group A with Ext1(A, Z) = 0 a free abelian group? The condition Ext1(A, Z) = 0 can be equivalently formulated as follows: whenever B is an abelian group and f : B → A is a surjective group... 1973 was a common year starting on Monday. ... Saharon Shelah (שהרן שלח, born July 3, 1945 in Jerusalem) is an Israeli mathematician. ... In axiomatic set theory, Martins axiom is a statement which is independent of the usual axioms of ZFC Set Theory. ... The axiom of constructibility is a possible axiom for set theory in mathematics. ...

Other

The answer to Suslin's problem is independent of ZFC. ◊ proves the existence of a Suslin line, while Martin's Axiom + the negation of the continuum hypothesis proves that no Suslin line exists. The other or constitutive other is a key concept in psychology and philosophy where it is often considered to be what defines or even constitutes the self (see self (psychology), self (philosophy), and self-concept) and other phenomena and cultural units: What appear to be cultural units--human beings, words... Suslins problem in mathematics is a question about orders posed by M. Suslin in the early 1920s. ... In mathematics, and particularly in axiomatic set theory, ◊S (diamondsuit or diamond) is a certain family of combinatorial principles. ... In axiomatic set theory, Martins axiom is a statement which is independent of the usual axioms of ZFC Set Theory. ...