Editing Cardinality of the continuum (section)

==The continuum hypothesis==
{{main|Continuum hypothesis}}

The continuum hypothesis asserts that <math>{\mathfrak c}</math> is also the second [[aleph number]], <math>\aleph_1</math>.<ref name=":0" /> In other words, the continuum hypothesis states that there is no set <math>A</math> whose cardinality lies strictly between <math>\aleph_0</math> and <math>{\mathfrak c}</math>
{{block indent|<math>\nexists A \quad:\quad \aleph_0 < |A| < \mathfrak c.</math>}}
This statement is now known to be independent of the axioms of [[Zermelo–Fraenkel set theory]] with the axiom of choice (ZFC), as shown by [[Kurt Gödel]] and [[Paul Cohen]].<ref>{{Cite book |last=Gödel |first=Kurt |date=1940-12-31 |title=Consistency of the Continuum Hypothesis. (AM-3) |url=http://dx.doi.org/10.1515/9781400881635 |doi=10.1515/9781400881635|isbn=9781400881635 }}</ref><ref>{{Cite journal |last=Cohen |first=Paul J. |title=The Independence of the Continuum Hypothesis |date=December 1963 |journal=Proceedings of the National Academy of Sciences |volume=50 |issue=6 |pages=1143–1148 |doi=10.1073/pnas.50.6.1143 |pmid=16578557 |pmc=221287 |bibcode=1963PNAS...50.1143C |issn=0027-8424|doi-access=free }}</ref><ref>{{Cite journal |last=Cohen |first=Paul J. |title=The Independence of the Continuum Hypothesis, Ii |date=January 1964 |journal=Proceedings of the National Academy of Sciences |volume=51 |issue=1 |pages=105–110 |doi=10.1073/pnas.51.1.105 |pmid=16591132 |pmc=300611 |bibcode=1964PNAS...51..105C |issn=0027-8424|doi-access=free }}</ref> That is, both the hypothesis and its negation are consistent with these axioms. In fact, for every nonzero [[natural number]] ''n'', the equality <math>{\mathfrak c}</math> = <math>\aleph_n</math> is independent of ZFC (case <math>n=1</math> being the continuum hypothesis). The same is true for most other alephs, although in some cases, equality can be ruled out by [[König's theorem (set theory)|König's theorem]] on the grounds of [[cofinality]] (e.g. <math>\mathfrak{c}\neq\aleph_\omega</math>). In particular, <math>\mathfrak{c}</math> could be either <math>\aleph_1</math> or <math>\aleph_{\omega_1}</math>, where <math>\omega_1</math> is the [[first uncountable ordinal]], so it could be either a [[successor cardinal]] or a [[limit cardinal]], and either a [[regular cardinal]] or a [[singular cardinal]].