List of statements undecidable in ZFC

List of statements undecidable in ZFC

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.

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 aleph_1, elements" is independent of ZFC.

Garth Dales and Robert M. Solovay proved in 1976 that Kaplansky'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.

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, the continuum hypothesis 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 omega_1. A similar example can be constructed using Martin's axiom. On the other hand, the consistency of the strong Fubini theorem was first showed by Friedman [Harvey Freidman, "A Consistent Fubini-Tonelli Theorem for Nonmeasureable Functions", Illinois J. Math., Vol. 24 (1980), no. 3, 390–395] . It can also be deduced from a variant of Freiling's axiom of symmetry [Chris Freiling, "Axioms of symmetry: throwing darts at the real number line", J. Symbolic Logic 51 (1986), no. 1, 190–200.] .

Axiomatic set theory

The consistency of ZFC was the first statement shown to be undecidable in ZFC.

The axiom "V=L" (that all sets are constructible) implies the generalized continuum hypothesis (which states that ℵ"n" = ℶn for every ordinal "n") and the combinatorial statement ◊, which both imply the continuum hypothesis (which states that 1 = 1). All these statements are independent of ZFC (as shown by Paul Cohen and Kurt Gödel).

Martin's axiom together with the negation of the continuum hypothesis is undecidable in ZFC.

The existence of large cardinal numbers, such as inaccessible cardinals, Mahlo cardinals etc., cannot be proved in ZFC, and few working set theorists expect them to be disproved. However it is not possible to formalize in ZFC a proof that ZFC cannot refute the existence of large cardinals (even under the added hypothesis that ZFC is itself consistent).

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 aleph_1 and 2^{aleph_0}). 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 2^{aleph_0}.

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.

Order theory

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.

References


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • List of mathematics articles (L) — NOTOC L L (complexity) L BFGS L² cohomology L function L game L notation L system L theory L Analyse des Infiniment Petits pour l Intelligence des Lignes Courbes L Hôpital s rule L(R) La Géométrie Labeled graph Labelled enumeration theorem Lack… …   Wikipedia

  • List of conjectures — This is an incomplete list of mathematical conjectures. They are divided into four sections, according to their status in 2007. See also: * Erdős conjecture, which lists conjectures of Paul Erdős and his collaborators * Unsolved problems in… …   Wikipedia

  • List of unsolved problems in mathematics — This article lists some unsolved problems in mathematics. See individual articles for details and sources. Contents 1 Millennium Prize Problems 2 Other still unsolved problems 2.1 Additive number theory …   Wikipedia

  • Undecidable problem — In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is impossible to construct an algorithm that leads to a yes or no answer the problem is not decidable.A decision problem is any …   Wikipedia

  • List of philosophy topics (R-Z) — RRaRabad Rabbinic law Rabbinic theology Francois Rabelais François Rabelais race racetrack paradox racism Gustav Radbruch Janet Radcliffe Richards Sarvepalli Radhakrishnan radical Aristotelianism radical behaviourism radical feminism radical… …   Wikipedia

  • Gödel's incompleteness theorems — In mathematical logic, Gödel s incompleteness theorems, proved by Kurt Gödel in 1931, are two theorems stating inherent limitations of all but the most trivial formal systems for arithmetic of mathematical interest. The theorems are of… …   Wikipedia

  • Zermelo–Fraenkel set theory — Zermelo–Fraenkel set theory, with the axiom of choice, commonly abbreviated ZFC, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics.ZFC consists of a single primitive ontological notion, that of… …   Wikipedia

  • Mathematical proof — In mathematics, a proof is a convincing demonstration (within the accepted standards of the field) that some mathematical statement is necessarily true.[1][2] Proofs are obtained from deductive reasoning, rather than from inductive or empirical… …   Wikipedia

  • Whitehead problem — In group theory, a branch of abstract algebra, the Whitehead problem is the following question::Is every abelian group A with Ext1( A , Z) = 0 a free abelian group?Abelian groups satisfying this condition are sometimes called Whitehead groups, so …   Wikipedia

  • Independence (mathematical logic) — In mathematical logic, a sentence sigma; is called independent of a given first order theory T if T neither proves nor refutes sigma;; that is, it is impossible to prove sigma; from T , and it is also impossible to prove from T that sigma; is… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”