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 ===
{{Multiple image
| direction         = horizontal
| image1            = Galileo Galilei (1564-1642) RMG BHC2700.tiff
| image2            = Bernard Bolzano.jpg
| total_width       = 350
| footer            = Portrait of [[Galileo Galilei]], circa 1640 (left).  Portrait of [[Bernard Bolzano]] 1781–1848 (right).
}}
[[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 ===
To better understand infinite sets, a notion of cardinality was formulated {{circa|1880}} by [[Georg Cantor]], the originator of [[set theory]]. He examined the process of equating two sets with a [[bijection]], a one-to-one correspondence between the elements of two sets. In 1891, with the publication of [[Cantor's diagonal argument|his diagonal argument]], he demonstrated that there are sets of numbers that cannot be placed in one-to-one correspondence with the set of natural numbers, i.e., there are "[[uncountable set]]s" that contain more elements than there are in the infinite set of natural numbers.<ref>{{cite journal |author=Georg Cantor |year=1891 |title=Ueber eine elementare Frage der Mannigfaltigkeitslehre |url=http://gdz.sub.uni-goettingen.de/pdfcache/PPN37721857X_0001/PPN37721857X_0001___LOG_0029.pdf |journal=Jahresbericht der Deutschen Mathematiker-Vereinigung |volume=1 |pages=75–78}}</ref>