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.
Abstract algebra
Charles Akemann
and
Nik Weaver
showed in
2003
that the statement "there exists a counterexample to
Naimark's problem
which is generated by
ℵ
1
elements" is independent of ZFC.
Axiomatic set theory
The
continuum hypothesis
(which states that
ℵ
1
=
ℶ
1
), 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 Gdel
), 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 Ext
1
(A,
Z
) = 0 a free abelian group?") is independent of ZFC, as shown in
1973
by
Saharon Shelah
.
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