Editing Cardinality (section)

===Cardinality of the continuum===
{{main|Cardinality of the continuum|Continuum hypothesis}}
[[File:Number line.png|thumb|The [[number line]], containing all points in its continuum.|314x314px]]
The [[number line]] is a geometric construct of the intuitive notions of "[[space]]" and "[[distance]]" wherein each point corresponds to a distinct quantity or position along a continuous path. The terms "continuum" and "continuous" refer to the totality of this line, having some space (other points) between any two points on the line ([[Dense order|dense]] and [[Archimedean property|archimedian]]) and the absence of any gaps ([[Completeness of the real numbers|completeness]]), This intuitive construct is formalized by the set of [[real numbers]] <math>(\R)</math> which model the continuum as a complete, densely ordered, uncountable set. 
[[File:Cantor set binary tree.svg|thumb|262x262px|First five itterations approaching the Cantor set]]
The [[Cardinality of the continuum|cardinality of the]] [[Cardinality of the continuum|continuum]], denoted by "<math>\mathfrak c</math>" (a lowercase [[fraktur (script)|fraktur script]] "c"), remains invariant under various transformations and mappings, many considered surprising. For example, all intervals on the real line e.g. <math>[0,1]</math>, and <math>[0,2]</math>, have the same cardinality as the entire set <math>\R</math>. First, <math>f(x) = 2x</math> is a bijection from <math>[0,1]</math> to <math>[0,2]</math>. Then, the [[tangent function]] is a bijection from the interval <math display="inline">\left( \frac{-\pi}{2} \, ,  \frac{\pi}{2} \right)</math> to the whole real line. A more surprising example is the [[Cantor set]], which is defined as follows: take the interval <math>[0,1]</math> and remove the middle third <math display="inline">\left( \frac{1}{3}, \frac{2}{3} \right)</math>, then remove the middle third of each of the two remaining segments, and continue removing middle thirds (see image). The Cantor set is the set of points that survive this process. This set that remains is all of the points whose decimal expansion can be written in [[Ternary numeral system|ternary]] without a 1. Reinterpreting these decimal expansions as [[Binary number|binary]] (e.g. by replacing the 2s with 1s) gives a bijection between the Cantor set and the interval <math>[0,1]</math>.
[[File:Peanocurve.svg|thumb|339x339px|Three iterations of a [[Peano curve]] construction, whose [[Limit of a sequence|limit]] is a [[space-filling curve]].]]
[[Space-filling curves]] are continuous surjective maps from the [[unit interval]] <math>[0,1]</math> onto the [[unit square]] on <math>\R^2</math>, with classical examples such as the [[Peano curve]] and [[Hilbert curve|Hilbert]] [[Hilbert curve|curve]]. Although such maps are not injective, they are indeed surjective, and thus suffice to demonstrate cardinal equivalence. They can be reused at each dimension to show that <math>|\R| = |\R^n| = \mathfrak{c}</math> for any dimension <math>n \geq 1.</math> The infinite [[cartesian product]] <math>\R^\infty</math>, can also be shown to have cardinality <math>\mathfrak c</math>. This can be established by cardinal exponentiation: <math>|\R^\infty| = \mathfrak{c}^{\aleph_0} = \left(2^{\aleph_0} \right)^{\aleph_0} = 2^{(\aleph_0 \cdot \aleph_0)} = 2^{\aleph_0} = \mathfrak{c} = |\R|</math>.  Thus, the real numbers, all finite-dimensional real spaces, and the countable cartesian product share the same cardinality.

As shown in {{slink||Unountable sets}}, the set of real numbers is strictly larger than the set of natural numbers. Specifically, <math>|\R| = |\mathcal{P}(\N)|  </math>. The [[Continuum Hypothesis]] (CH) asserts that the real numbers have the next largest cardinality after the natural numbers, that is <math>|\R| = \aleph_1</math>. As shown by [[Kurt Gödel|Gödel]] and [[Paul Cohen|Cohen]], 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 [[Consistency|consistent]].<ref>{{Cite journal |last=Cohen |first=Paul J. |date=December 15, 1963 |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 |pages=1143–1148 |bibcode=1963PNAS...50.1143C |doi=10.1073/pnas.50.6.1143 |jstor=71858 |pmc=221287 |pmid=16578557 |doi-access=free}}</ref><ref>{{Cite journal |last=Cohen |first=Paul J. |date=January 15, 1964 |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 |pages=105–110 |bibcode=1964PNAS...51..105C |doi=10.1073/pnas.51.1.105 |jstor=72252 |pmc=300611 |pmid=16591132 |doi-access=free}}</ref><ref>{{Citation |last=Penrose |first=R |title=The Road to Reality: A Complete Guide to the Laws of the Universe |year=2005 |publisher=Vintage Books |isbn=0-09-944068-7 |author-link=Roger Penrose}}</ref> The [[Generalized Continuum Hypothesis]] (GCH) extends this to all infinite cardinals, stating that <math>2^{\aleph_\alpha} = \aleph_{\alpha+1}</math> for every ordinal <math>\alpha</math>. Without GHC, the cardinality of <math>\R</math> cannot be written in terms of alephs. The [[Beth numbers]] provide a concise notation for powersets of the real numbers starting from <math>\beth_1 = |\R|</math>, then <math>\beth_2 = |\mathcal{P}(\R)| = 2^{\beth_1}</math>, and in general <math>\beth_{n+1} = 2^{\beth_n}</math>.