Editing Continuum hypothesis (section)

==Independence from ZFC==

The independence of the continuum hypothesis (CH) from [[Zermelo–Fraenkel set theory]] (ZF) follows from combined work of [[Kurt Gödel]]  and [[Paul Cohen (mathematician)|Paul Cohen]].

Gödel<ref>

{{cite journal|doi=10.1073/pnas.24.12.556 |title=The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis |date=1938 |last1=Gödel |first1=Kurt |journal=Proceedings of the National Academy of Sciences |volume=24 |issue=12 |pages=556–557 |pmid=16577857 |pmc=1077160 |bibcode=1938PNAS...24..556G |doi-access=free }}

</ref>{{r|Gödel1940}}
showed that CH cannot be disproved from ZF, even if the [[axiom of choice]] (AC) is adopted, i.e. from ZFC. Gödel's proof shows that both CH and AC hold in the [[constructible universe]] <math>L</math>, an [[inner model]] of ZF set theory, assuming only the axioms of ZF.  The existence of an inner model of ZF in which additional axioms hold shows that the additional axioms are (relatively) [[consistent]] with ZF, provided ZF itself is consistent. The latter condition cannot be proved in ZF itself, due to [[Gödel's incompleteness theorems]], but is widely believed to be true and can be proved in stronger set theories.

Cohen{{r|Cohen1963|Cohen1964}} showed that CH cannot be proven from the ZFC axioms, completing the overall independence proof. To prove his result, Cohen developed the method of [[Forcing (mathematics)|forcing]], which has become a standard tool in set theory. Essentially, this method begins with a model of ZF in which CH holds and constructs another model which contains more sets than the original in a way that CH does not hold in the new model. Cohen was awarded the [[Fields Medal]] in 1966 for his proof.

Cohen's independence proof shows that CH is independent of ZFC. Further research has shown that CH is independent of all known ''[[large cardinal axiom]]s'' in the context of ZFC.{{r|Feferman1999_99111}} Moreover, it has been shown that the [[cardinality of the continuum]] <math>\mathfrak c = 2^{\aleph_0}</math> can be any cardinal consistent with [[Kőnig's theorem (set theory)|Kőnig's theorem]]. A result of Solovay, proved shortly after Cohen's result on the independence of the continuum hypothesis, shows that in any model of ZFC, if <math>\kappa</math> is a cardinal of uncountable [[cofinality]], then there is a forcing extension in which <math>2^{\aleph_0} = \kappa</math>. However, per Kőnig's theorem, it is not consistent to assume <math>2^{\aleph_0}</math> is <math>\aleph_\omega</math> or <math>\aleph_{\omega_1+\omega}</math> or any cardinal with cofinality <math>\omega</math>.

The continuum hypothesis is closely related to [[Statements Equivalent to the Continuum Hypothesis|many statements]] in [[mathematical analysis|analysis]], point set [[topology]] and [[measure theory]]. As a result of its independence, many substantial [[conjecture]]s in those fields have subsequently been shown to be independent as well.

The independence from ZFC means that proving or disproving the CH within ZFC is impossible.  However, Gödel and Cohen's negative results are not universally accepted as disposing of all interest in the continuum hypothesis. The continuum hypothesis remains an active topic of research: see [[W. Hugh Woodin|Woodin]]{{r|Woodin2001a|Woodin2001b}} and [[Peter Koellner|Koellner]]{{r|Koellner2011a}} for an overview of the current research status.

The continuum hypothesis and the [[axiom of choice]] were among the first genuinely mathematical statements shown to be independent of ZF set theory. Although the existence of some statements independent of ZFC had already been known more than two decades prior: for example, assuming [[ω-consistency|good soundness properties]] and the consistency of ZFC, [[Gödel's incompleteness theorems]] published in 1931 establish that there is a formal statement Con(ZFC) (one for each appropriate [[Gödel numbering]] scheme) expressing the consistency of ZFC, that is also independent of it. The latter independence result indeed holds for many theories.