Editing Cardinality (section)

===Cardinality of the continuum===
{{main|Cardinality of the continuum}}
[[File:Number line.png|thumb|The [[number line]], containing all points in its continuum.]]
The cardinality of the [[real number]]s is denoted by "<math>\mathfrak c</math>" (a lowercase [[fraktur (script)|fraktur script]] "c"), and is also referred to as the [[cardinality of the continuum]]. Cantor showed, using the [[Cantor's diagonal argument|diagonal argument]], that <math>{\mathfrak c} >\aleph_0.</math> We can show that <math>\mathfrak c = 2^{\aleph_0},</math> this also being the cardinality of the set of all subsets of the natural numbers.

The [[continuum hypothesis]] says that <math>\aleph_1 = 2^{\aleph_0},</math> i.e. <math>2^{\aleph_0}</math> is the smallest cardinal number bigger than <math>\aleph_0,</math> i.e. there is no set whose cardinality is strictly between that of the integers and that of the real numbers. The continuum hypothesis is [[independence (mathematical logic)|independent]] of [[Zermelo–Fraenkel set theory with the axiom of choice|ZFC]], a standard axiomatization of set theory; that is, it is impossible to prove the continuum hypothesis or its negation from ZFC—provided that ZFC is consistent.<ref>{{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, 1963 | pages = 1143–1148
 | doi = 10.1073/pnas.50.6.1143
 | pmid = 16578557
 | pmc = 221287 | jstor=71858
 | bibcode = 1963PNAS...50.1143C
 | doi-access = free
 }}</ref><ref>{{Cite journal
 | first = Paul J. | last = Cohen
 | title = The Independence of the Continuum Hypothesis, II
 | journal = Proceedings of the National Academy of Sciences of the United States of America
 | volume = 51 | issue = 1 | date = January 15, 1964 | pages = 105–110
 | doi = 10.1073/pnas.51.1.105
 | pmid = 16591132
 | pmc = 300611 | jstor=72252
 | bibcode = 1964PNAS...51..105C
 | doi-access = free
 }}</ref><ref>{{Citation|first=R|last=Penrose|author-link=Roger Penrose|title=The Road to Reality: A Complete Guide to the Laws of the Universe|publisher=Vintage Books|year=2005|isbn=0-09-944068-7}}</ref>

One of Cantor's most important results was that the [[cardinality of the continuum]] (<math>\mathfrak{c}</math>) is greater than that of the natural numbers (<math>\aleph_0</math>); that is, there are more real numbers '''R''' than natural numbers '''N'''. Namely, Cantor showed that <math>\mathfrak{c} = 2^{\aleph_0} = \beth_1</math> (see [[Beth number#Beth one|Beth one]]) satisfies:
:<math>2^{\aleph_0} > \aleph_0</math>
:(see [[Cantor's diagonal argument]] or [[Cantor's first uncountability proof]]).

The [[continuum hypothesis]] states that there is no [[cardinal number]] between the cardinality of the reals and the cardinality of the natural numbers, that is,
:<math>2^{\aleph_0} = \aleph_1</math>

However, this hypothesis can neither be proved nor disproved within the widely accepted [[ZFC]] [[axiomatic set theory]], if ZFC is consistent.

The first of these results is apparent by considering, for instance, the [[tangent function]], which provides a [[one-to-one correspondence]] between the [[Interval (mathematics)|interval]] (−{{sfrac|1|2}}π, {{sfrac|1|2}}π) and '''R'''.

The second result was first demonstrated by Cantor in 1878, but it became more apparent in 1890, when [[Giuseppe Peano]] introduced the [[space-filling curve]]s, curved lines that twist and turn enough to fill the whole of any square, or cube, or [[hypercube]], or finite-dimensional space. These curves are not a direct proof that a line has the same number of points as a finite-dimensional space, but they can be used to obtain [[Space-filling curve#Proof that a square and its side contain the same number of points|such a proof]].

Cantor also showed that sets with cardinality strictly greater than <math>\mathfrak c</math> exist (see his [[Cantor's diagonal argument#General sets|generalized diagonal argument]] and [[Cantor's theorem|theorem]]). They include, for instance:

:* the set of all subsets of '''R''', i.e., the [[power set]] of '''R''', written ''P''('''R''') or 2<sup>'''R'''</sup>
:* the set '''R'''<sup>'''R'''</sup> of all functions from '''R''' to '''R'''

Both have cardinality
:<math>2^\mathfrak {c} = \beth_2 > \mathfrak c </math>
:(see [[Beth number#Beth two|Beth two]]).

The [[Cardinality of the continuum#Cardinal equalities|cardinal equalities]] <math>\mathfrak{c}^2 = \mathfrak{c},</math> <math>\mathfrak c^{\aleph_0} = \mathfrak c,</math> and <math>\mathfrak c ^{\mathfrak c} = 2^{\mathfrak c}</math> can be demonstrated using [[cardinal arithmetic]]:
:<math>\mathfrak{c}^2 = \left(2^{\aleph_0}\right)^2 = 2^{2\cdot{\aleph_0}} = 2^{\aleph_0} = \mathfrak{c},</math>
:<math>\mathfrak c^{\aleph_0} = \left(2^{\aleph_0}\right)^{\aleph_0} = 2^{{\aleph_0}\cdot{\aleph_0}} = 2^{\aleph_0} = \mathfrak{c},</math>
:<math> \mathfrak c ^{\mathfrak c} = \left(2^{\aleph_0}\right)^{\mathfrak c} = 2^{\mathfrak c\cdot\aleph_0} = 2^{\mathfrak c}.</math>