Editing Cardinality (section)

==History==

=== Prehistory ===
A crude sense of cardinality, an awareness that groups of things or events compare with other groups by containing more, fewer, or the same number of instances, is observed in a variety of present-day animal species, suggesting an origin millions of years ago.<ref>Cepelewicz, Jordana   ''[https://www.quantamagazine.org/animals-can-count-and-use-zero-how-far-does-their-number-sense-go-20210809/ Animals Count and Use Zero. How Far Does Their Number Sense Go?]'', [[Quanta Magazine|Quanta]], August 9, 2021</ref> Human expression of cardinality is seen as early as {{val|40000}} years ago, with equating the size of a group with a group of recorded notches, or a representative collection of other things, such as sticks and shells.<ref>{{Cite web|url=https://mathtimeline.weebly.com/early-human-counting-tools.html|title=Early Human Counting Tools|website=Math Timeline|access-date=2018-04-26}}</ref> The abstraction of cardinality as a number is evident by 3000 BCE, in Sumerian [[History of mathematics|mathematics]] and the manipulation of numbers without reference to a specific group of things or events.<ref>Duncan J. Melville (2003). [http://it.stlawu.edu/~dmelvill/mesomath/3Mill/chronology.html Third Millennium Chronology] {{Webarchive|url=https://web.archive.org/web/20180707213616/http://it.stlawu.edu/~dmelvill/mesomath/3Mill/chronology.html |date=2018-07-07 }}, ''Third Millennium Mathematics''. [[St. Lawrence University]].</ref>

=== Ancient history ===
[[File:AristotlesWheelLabeledDiagram.svg|thumb|252x252px|Diagram of Aristotle's wheel as described in ''Mechanica''.]]
From the 6th century BCE, the writings of Greek philosophers show hints of infinite cardinality. While they considered generally infinity as an endless series of actions, such as adding 1 to a number repeatedly, they considered rarely infinite sets ([[actual infinity]]), and, if they did, they considered infinity as a unique cardinality.<ref name="Allen2">{{Cite web |last=Allen |first=Donald |date=2003 |title=The History of Infinity |url=https://www.math.tamu.edu/~dallen/masters/infinity/infinity.pdf |url-status=dead |archive-url=https://web.archive.org/web/20200801202539/https://www.math.tamu.edu/~dallen/masters/infinity/infinity.pdf |archive-date=August 1, 2020 |access-date=Nov 15, 2019 |website=Texas A&M Mathematics}}</ref> The ancient Greek notion of infinity also considered the division of things into parts repeated without limit. 

One of the earliest explicit uses of a one-to-one correspondence is recorded in [[Aristotle]]'s [[Mechanics (Aristotle)|''Mechanics'']] ({{Circa|350 BC}}), known as [[Aristotle's wheel paradox]]. The paradox can be briefly described as follows: A wheel is depicted as two [[concentric circles]]. The larger, outer circle is tangent to a horizontal line (e.g. a road that it rolls on), while the smaller, inner circle is rigidly affixed to the larger.  Assuming the larger circle rolls along the line without slipping (or skidding) for one full revolution, the distances moved by both circles are the same: the [[circumference]] of the larger circle. Further, the lines traced by the bottom-most point of each is the same length.<ref name=":0">{{Cite journal |last=Drabkin |first=Israel E. |date=1950 |title=Aristotle's Wheel: Notes on the History of a Paradox |journal=Osiris |volume=9 |pages=162–198 |doi=10.1086/368528 |jstor=301848 |s2cid=144387607}}</ref> Since the smaller wheel does not skip any points, and no point on the smaller wheel is used more than once, there is a one-to-one correspondence between the two circles.

=== Pre-Cantorian set theory ===
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[[Galileo Galilei]] presented what was later coined [[Galileo's paradox]] in his book ''[[Two New Sciences]]'' (1638), where he attempts to show that infinite quantities cannot be called greater or less than one another. He presents the paradox roughly as follows: a [[square number]] is one which is the product of another number with itself, such as 4 and 9, which are the squares of 2 and 3, respectively. Then the [[square root]] of a square number is that multiplicand. He then notes that there are as many square numbers as there are square roots, since every square has its own root and every root its own square, while no square has more than one root and no root more than one square. But there are as many square roots as there are numbers, since every number is the square root of some square. He, however, concluded that this meant we could not compare the sizes of infinite sets, missing the opportunity to discover cardinality.<ref>{{Cite book |last=Galilei |first=Galileo |author-link=Galileo Galilei |url=https://dn790007.ca.archive.org/0/items/dialoguesconcern00galiuoft/dialoguesconcern00galiuoft.pdf |title=Dialogues Concerning Two New Sciences |publisher=[[The Macmillan Company]] |year=1914 |location=New York |pages=31–33 |language=en |translator-last=Crew |translator-first=Henry |orig-year=1638 |translator-last2=De Salvio |translator-first2=Alfonso}}</ref>

[[Bernard Bolzano]]'s ''[[Paradoxes of the Infinite]]'' (''Paradoxien des Unendlichen'', 1851) is often considered the first systematic attempt to introduce the concept of sets into [[mathematical analysis]]. In this work, Bolzano defended the notion of [[actual infinity]], examined various properties of infinite collections, including an early formulation of what would later be recognized as one-to-one correspondence between infinite sets, and proposed to base mathematics on a notion similar to sets. He discussed examples such as the pairing between the [[Interval (mathematics)|intervals]] <math>[0,5]</math> and <math>[0,12]</math> by the relation <math>5y =  12x.</math> Bolzano also revisited and extended Galileo's paradox. However, he too resisted saying that these sets were, in that sense, the same size. Thus, while ''Paradoxes of the Infinite'' anticipated several ideas central to later set theory, the work had little influence on contemporary mathematics, in part due to its [[posthumous publication]] and limited circulation.<ref>{{Citation |last=Ferreirós |first=José |title=The Early Development of Set Theory |date=2024 |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/settheory-early/ |access-date=2025-01-04 |archive-url=https://web.archive.org/web/20210512135148/https://plato.stanford.edu/entries/settheory-early/ |archive-date=2021-05-12 |url-status=live |edition=Winter 2024 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri |encyclopedia=The Stanford Encyclopedia of Philosophy}}</ref><ref>{{Citation |last=Bolzano |first=Bernard |title=Einleitung zur Größenlehre und erste Begriffe der allgemeinen Größenlehre |volume=II, A, 7 |page=152 |year=1975 |editor-last=Berg |editor-first=Jan |series=Bernard-Bolzano-Gesamtausgabe, edited by Eduard Winter et al. |location=Stuttgart, Bad Cannstatt |publisher=Friedrich Frommann Verlag |isbn=3-7728-0466-7 |author-link=Bernard Bolzano}}</ref><ref>{{Cite book |last=Bolzano |first=Bernard |url=https://archive.org/details/dli.ernet.503861/ |title=Paradoxes Of The Infinite |date=1950 |publisher=Routledge and Kegan Paul |location=London |translator-last=Prihonsky |translator-first=Fr.}}</ref>

Other, more minor contributions incude [[David Hume]] in ''[[A Treatise of Human Nature]]'' (1739), who said ''"When two numbers are so combined, as that the one has always a unit answering to every unit of the other, we pronounce them equal",<ref>{{cite book |last=Hume |first=David |date=1739–1740 |title=A Treatise of Human Nature |chapter=Part III. Of Knowledge and Probability: Sect. I. Of Knowledge |chapter-url=https://gutenberg.org/cache/epub/4705/pg4705-images.html#link2H_4_0021 |via=Project Gutenberg}}</ref>'' now called ''[[Hume's principle]]'', which was used extensively by [[Gottlob Frege]] later during the rise of set theory.<ref>{{cite book |last=Frege |first=Gottlob |date=1884 |title=Die Grundlagen der Arithmetik |chapter=IV. Der Begriff der Anzahl § 63. Die Möglichkeit der eindeutigen Zuordnung als solches. Logisches Bedenken, dass die Gleichheit für diesen Fall besonders erklärt wird |quote=§63. Ein solches Mittel nennt schon Hume: »Wenn zwei Zahlen so combinirt werden, dass die eine immer eine Einheit hat, die jeder Einheit der andern entspricht, so geben wir sie als gleich an.« |chapter-url=https://gutenberg.org/cache/epub/48312/pg48312-images.html#para_63 |via=Project Gutenberg}}</ref> [[Jakob Steiner]], whom [[Georg Cantor]] credits the original term, ''Mächtigkeit'', for cardinality (1867).<ref name=":2" /><ref name=":3" /><ref name=":4" /> [[Peter Gustav Lejeune Dirichlet]] is commonly credited for being the first to explicitly formulate the [[pigeonhole principle]] in 1834,<ref>Jeff Miller, Peter Flor, Gunnar Berg, and Julio González Cabillón. "[http://jeff560.tripod.com/p.html Pigeonhole principle]". In Jeff Miller (ed.) ''[http://jeff560.tripod.com/mathword.html Earliest Known Uses of Some of the Words of Mathematics]''. Electronic document, retrieved November 11, 2006</ref> though it was used at least two centuries earlier by [[Jean Leurechon]] in 1624.<ref name="leurechon">{{cite journal |last1=Rittaud |first1=Benoît |last2=Heeffer |first2=Albrecht |year=2014 |title=The pigeonhole principle, two centuries before Dirichlet |url=https://biblio.ugent.be/publication/4115264 |journal=The Mathematical Intelligencer |volume=36 |issue=2 |pages=27–29 |doi=10.1007/s00283-013-9389-1 |mr=3207654 |s2cid=44193229 |hdl-access=free |hdl=1854/LU-4115264}}</ref>

=== 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>