Science Fair Project Encyclopedia
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 ℵ1 elements" is independent of ZFC.
Axiomatic set theory
The continuum hypothesis (which states that ℵ1 = ℶ1), and the generalized continuum hypothesis (which states that ℵn = ℶn for every n) are independent of ZFC (as shown by Paul Cohen and Kurt Gödel), as is the combinatorial statement ◊ (which implies CH).
The existence of large cardinal numbers, such as inaccessible cardinals, Mahlo cardinals etc., can neither be proven nor disproven in ZFC.
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.
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details