Infinity Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.Anti-spam check. Do not fill this in! ====Cardinality of the continuum==== {{Main|Cardinality of the continuum}} One of Cantor's most important results was that the cardinality of the continuum <math>\mathbf c</math> is greater than that of the natural numbers <math>{\aleph_0}</math>; that is, there are more real numbers {{math|'''R'''}} than natural numbers {{math|'''N'''}}. Namely, Cantor showed that <math>\mathbf{c}=2^{\aleph_0}>{\aleph_0}</math>.<ref>{{Cite journal| last = Dauben | first = Joseph | title = Georg Cantor and the Battle for Transfinite Set Theory | url = http://acmsonline.org/home2/wp-content/uploads/2016/05/Dauben-Cantor.pdf | journal = 9th ACMS Conference Proceedings | year = 1993 | page = 4 }}</ref>{{further|Cantor's diagonal argument|Cantor's first set theory article}} 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>\mathbf{c}=\aleph_1=\beth_1</math>.{{further|Beth number#Beth one}}This hypothesis cannot be proved or disproved within the widely accepted [[Zermelo–Fraenkel set theory]], even assuming the [[Axiom of Choice]].<ref>{{harvnb|Cohen|1963|p=1143}}</ref> [[Cardinal arithmetic]] can be used to show not only that the number of points in a [[real number line]] is equal to the number of points in any [[line segment|segment of that line]], but also that this is equal to the number of points on a plane and, indeed, in any [[finite-dimensional]] space.{{citation needed|date=April 2017}} [[File:Peanocurve.svg|thumb|The first three steps of a fractal construction whose limit is a [[space-filling curve]], showing that there are as many points in a one-dimensional line as in a two-dimensional square]] The first of these results is apparent by considering, for instance, the [[tangent (trigonometric function)|tangent]] function, which provides a [[one-to-one correspondence]] between the [[Interval (mathematics)|interval]] ({{math|−{{sfrac|π|2}}, {{sfrac|π|2}}}}) and{{math| '''R'''}}.{{see also|Hilbert's paradox of the Grand Hotel}}The second result was proved by Cantor in 1878, but only became intuitively 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 can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.<ref>{{harvnb|Sagan|1994|pp=10–12}}</ref> Summary: Please note that all contributions to Christianpedia may be edited, altered, or removed by other contributors. If you do not want your writing to be edited mercilessly, then do not submit it here. You are also promising us that you wrote this yourself, or copied it from a public domain or similar free resource (see Christianpedia:Copyrights for details). Do not submit copyrighted work without permission! Cancel Editing help (opens in new window) Discuss this page