Editing Cardinality (section)

=== Early set theory ===

==== 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]].

==== Other contributors ====
Parallel to Cantor’s development, [[Richard Dedekind]] independently formulated [[Dedekind-infinite set|a definition of infinite set]] as one that can be placed in bijection with a proper subset of itself, which was shown to be equivalent with Cantor’s definition of cardinality (given the [[axiom of choice]]). Dedekind’s ''[[Was sind und was sollen die Zahlen?]]'' (1888) emphasized structural properties over extensional definitions, and supported the bijective formulation of size and number. Dedekind was in correspondence with Cantor during the development of set theory; he supplied Cantor with a proof of the countability of the [[algebraic numbers]], and gave feedback and modifications on Cantor's proofs before publishing.

After Cantor's 1883 proof that all finite-dimensional [[manifolds]] have the same cardinality,<ref>{{Cite journal |last=Cantor |first=Georg |date=1883-12-01 |title=Ueber unendliche, lineare Punktmannichfaltigkeiten |url=https://doi.org/10.1007/BF01446819 |journal=Mathematische Annalen |language=de |volume=21 |issue=4 |pages=545–591 |doi=10.1007/BF01446819 |issn=1432-1807}}</ref>{{clarify|reason=Cantor dis not know the modern notion of a manifols. Using "manifold" here seem a mistranslation.|date=June 2025}} in 1890, [[Giuseppe Peano]] introducted the [[Peano curve]], which  was a more visual proof that the [[unit interval]] <math>[0,1]</math> has the same cardinality as the [[unit square]] on <math>\R^2.</math><ref>{{Cite journal |last=Peano |first=G. |date=1890-03-01 |title=Sur une courbe, qui remplit toute une aire plane |url=https://doi.org/10.1007/BF01199438 |journal=Mathematische Annalen |language=fr |volume=36 |issue=1 |pages=157–160 |doi=10.1007/BF01199438 |issn=1432-1807 |archive-url=https://archive.org/details/PeanoSurUneCurve |archive-date=2018-07-22}}</ref> This created a new area of mathematical analysis studying what is now called [[space-filling curves]].<ref>{{citation |last=Gugenheimer |first=Heinrich Walter |title=Differential Geometry |page=3 |year=1963 |url=https://books.google.com/books?id=CSYtkV4NTioC&pg=PA |publisher=Courier Dover Publications |isbn=9780486157207}}.</ref> 

German logician [[Gottlob Frege]] sought to ground the concept of number in logic, defining numbers using Cantor's theory of cardinality, connecting the notion to [[Hume's principle]]. In ''[[Die Grundlagen der Arithmetik]]'' (1884) and the subsequent ''Grundgesetze der Arithmetik'' (1893, 1903), Frege attempted to derive arithmetic from logical principles, treating cardinality and cardinal number as a [[primitive notion]]. However, Frege's approach to set theory was undermined by the discovery of [[Russell's paradox]] in 1901. The paradox played a crucial role in the [[foundational crisis in mathematics]] and especially the [[Logicism#History|logicist program]]. 

This was eventually resolved by [[Bertrand Russell]] himself in ''[[Principia Mathematica]]'' (1910{{En dash}}1913, vol. II),{{Sfn|Russell|Whitehead}} co-authored with [[Alfred North Whitehead]], which introduced a [[Type theory#History|theory of types]] to avoid such paradoxes, defining cardinal numbers at each level of the type hierarchy. Cardinal numbers were treated as [[equivalence classes]] of sets under equinumerosity, but only within a type-theoretic framework. Though Russell initially had difficulties understanding Cantor's and Frege’s intuitions of cardinality, shown in his 1905 manuscript ''On Some Difficulties in the Theory of Transfinite Numbers and Order Types.''<ref>{{Cite journal |last=Russell |first=B. |date=1907 |title=On Some Difficulties in the Theory of Transfinite Numbers and Order Types |url=https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s2-4.1.29?doi=10.1112%2Fplms%2Fs2-4.1.29 |journal=Proceedings of the London Mathematical Society |language=en |volume=s2-4 |issue=1 |pages=29–53 |doi=10.1112/plms/s2-4.1.29 |issn=1460-244X}}</ref>