Editing Cardinality (section)

=== 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>