Editing Cardinality (section)

==== Georg Cantor ====
[[File:Georg_Cantor3.jpg|alt=refer to caption|thumb|339x339px|[[Georg Cantor]], {{spaces|4|hair}}{{circa}} 1870]]
The concept of cardinality, as a formal measure of the size of a set, emerged nearly fully formed in the work of Georg Cantor during the 1870s and 1880s, in the context of [[mathematical analysis]]. In a series of papers beginning with ''[[Cantor's first set theory article|On a Property of the Collection of All Real Algebraic Numbers]]'' (1874),<ref>{{Citation |last=Cantor |first=Herrn |title=Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen |date=1984 |work=Über unendliche, lineare Punktmannigfaltigkeiten: Arbeiten zur Mengenlehre aus den Jahren 1872–1884 |pages=19–24 |editor-last=Cantor |editor-first=Georg |orig-date=1874 |url=https://link.springer.com/chapter/10.1007/978-3-7091-9516-1_2 |access-date=2025-05-24 |place=Vienna |publisher=Springer |language=de |doi=10.1007/978-3-7091-9516-1_2 |isbn=978-3-7091-9516-1}}</ref> Cantor introduced the idea of comparing the sizes of infinite sets, through the notion of one-to-one correspondence. He showed that the set of [[real numbers]] was, in this sense, strictly larger than the set of natural numbers [[Cantor's first set theory article#Second theorem|using a nested intervals argument]]. This result was later refined into the more widely known [[Cantor's diagonal argument|diagonal argument]] of 1891, published in ''Über eine elementare Frage der Mannigfaltigkeitslehre,''<ref>{{Cite journal |last=Cantor |first=Georg |date=1890 |title=Ueber eine elementare Frage der Mannigfaltigketislehre. |url=https://eudml.org/doc/144383 |journal=Jahresbericht der Deutschen Mathematiker-Vereinigung |volume=1 |pages=72–78 |issn=0012-0456}}</ref> where he also proved the more general result (now called [[Cantor's Theorem]]) that the [[power set]] of any set is strictly larger than the set itself. 

Cantor introduced the notion [[cardinal numbers]] in terms of [[ordinal numbers]]. He viewed cardinal numbers as an abstraction of sets, introduced the notations, where, for a given set <math display="inline">M</math>, the [[order type]] of that set was written <math display="inline">\overline{M}</math>, and the cardinal number was <span style="border-top: 3px double;"><math display="inline">M</math></span>, a double abstraction. He also introduced the [[Cardinality#Aleph numbers|Aleph sequence]] for infinite cardinal numbers. These notations appeared in correspondence and were formalized in his later writings, particularly the series ''Beiträge zur Begründung der transfiniten Mengenlehre'' (1895{{En dash}}1897).<ref>{{Cite journal |last=Cantor |first=Georg |date=1895-11-01 |title=Beiträge zur Begründung der transfiniten Mengenlehre |url=https://link.springer.com/article/10.1007/BF02124929 |journal=Mathematische Annalen |language=de |volume=46 |issue=4 |pages=481–512 |doi=10.1007/BF02124929 |issn=1432-1807}}</ref> In these works, Cantor developed an [[Cardinal arithmetic|arithmetic of cardinal numbers]], defining addition, multiplication, and exponentiation of cardinal numbers based on set-theoretic constructions.  This led to the formulation of the [[Continuum Hypothesis]] (CH), the proposition that no set has cardinality strictly between <math>\aleph_0</math> and the [[cardinality of the continuum]], that is <math>|\R| = \aleph_1</math>. Cantor was unable to resolve CH and left it as an [[open problem]].