Editing Set (mathematics) (section)

===Cardinality of the real numbers===

The cardinality of set of the [[real numbers]] is called the [[cardinality of the continuum]] and denoted {{tmath|\mathfrak c}}. (The term "[[continuum (set theory)|continuum]]" referred to the [[real line]] before the 20th century, when the real line was not commonly viewed as a set of numbers.)

Since, as seen above, the real line {{tmath|\R}} has the same cardinality of an [[open interval]], every subset of {{tmath|\R}} that contains a nonempty [[open interval]] has also the cardinality {{tmath|\mathfrak c}}.

One has 
<math display=block>\mathfrak c = 2^{\aleph_0},</math>
meaning that the cardinality of the real numbers equals the cardinality of the [[power set]] of the natural numbers. In particular,<ref name="Stillwell2013">{{cite book|author=John Stillwell|title=The Real Numbers: An Introduction to Set Theory and Analysis|url=https://books.google.com/books?id=VPe8BAAAQBAJ|date=16 October 2013|publisher=Springer Science & Business Media|isbn=978-3-319-01577-4}}</ref>   
<math display=block>\mathfrak c > \aleph_0.</math>

When published in 1878 by [[Georg Cantor]],<ref name = "Cantor1878" /> this result was so astonishing that it was refused by mathematicians, and several tens years were needed before its common acceptance.

It can be shown that {{tmath|\mathfrak c}} is also the cardinality of the entire [[plane (mathematics)|plane]], and of any [[dimension (mathematics)|finite-dimensional]] [[Euclidean space]].<ref name="Tall2006">{{cite book|author=David Tall|title=Advanced Mathematical Thinking|url=https://books.google.com/books?id=czKqBgAAQBAJ&pg=PA212|date=11 April 2006|publisher=Springer Science & Business Media|isbn=978-0-306-47203-9|pages=211}}</ref>

The [[continuum hypothesis]], was a conjecture formulated by Georg Cantor in 1878 that there is no set with cardinality strictly between {{tmath|\aleph_0}} and {{tmath|\mathfrak c}}.<ref name = "Cantor1878">{{Cite journal
 | first = Georg | last = Cantor
 | title = Ein Beitrag zur Mannigfaltigkeitslehre
 | journal = [[Journal für die Reine und Angewandte Mathematik]]
 | volume = 1878 | issue = 84
 | year = 1878 | pages=242–258
 | url = http://www.digizeitschriften.de/dms/img/?PPN=PPN243919689_0084&DMDID=dmdlog15 | doi=10.1515/crll.1878.84.242| doi-broken-date = 1 November 2024
 }}</ref> In 1963, [[Paul Cohen]] proved that the continuum hypothesis is [[independence (mathematical logic)|independent]] of the [[axiom]]s of [[Zermelo–Fraenkel set theory]] with the [[axiom of choice]].<ref name = "Cohen1963a">
{{Cite journal
 | first = Paul J. | last = Cohen
 | title = The Independence of the Continuum Hypothesis
 | journal = Proceedings of the National Academy of Sciences of the United States of America
 | volume = 50 | issue = 6 | date = December 15, 1963a | pages = 1143–1148
 | doi = 10.1073/pnas.50.6.1143
 | pmid = 16578557
 | pmc = 221287 | jstor=71858
 | bibcode = 1963PNAS...50.1143C| doi-access = free
 }}
</ref> 
This means that if the most widely used [[set theory]] is [[consistency|consistent]] (that is not self-contradictory),{{efn|The consistency of set theory cannot proved from within itself.}} then the same is true for both the set theory with the continuum hypothesis added as a further axiom, and the set theory with the negation of the continuum hypothesis added.