Editing Cardinality (section)

== Etymology ==
In English, the term ''cardinality'' originates from the [[Post-Classical Latin language|post-classical Latin]] ''cardinalis'', meaning "principal" or "chief", which derives from ''cardo'', a noun meaning "hinge". In Latin, ''cardo'' referred to something central or pivotal, both literally and metaphorically. This concept of centrality passed into [[medieval Latin]] and then into English, where ''cardinal'' came to describe things considered to be, in some sense, fundamental, such as ''[[cardinal virtues]]'', ''[[cardinal sins]]'', ''[[cardinal directions]]'', and (in the grammatical sense) ''[[Cardinal numbers (linguistics)|cardinal numbers]].<ref>[[Oxford English Dictionary]], "cardinal (adj.), Etymology," March 2025, https://doi.org/10.1093/OED/1490074521.</ref>''<ref>Harper Douglas, "Etymology of cardinal," [[Etymonline]], Online Etymology Dictionary, accessed April 20, 2025, https://www.etymonline.com/word/cardinal.</ref> The last of which referred to numbers used for counting (e.g., one, two, three),<ref>''[[Oxford English Dictionary]]'', "cardinal number (''n.''), sense 1," July 2023, https://doi.org/10.1093/OED/3193437451.</ref> as opposed to ''[[Ordinal numbers (linguistics)|ordinal numbers]]'', which express order (e.g., first, second, third),<ref>[[Oxford English Dictionary]], "ordinal (n.2)," June 2024, https://doi.org/10.1093/OED/6032173309.</ref> and [[Nominal number|''nominal numbers'']] used for labeling without meaning (e.g., [[jersey numbers]] and [[serial numbers]]).<ref>{{cite journal |last1=Woodin |first1=Greg |last2=Winter |first2=Bodo |year=2024 |title=Numbers in Context: Cardinals, Ordinals, and Nominals in American English |journal=Cognitive Science |volume=48 |doi=10.1111/cogs.13471 |pmc=11475258 |pmid=38895756 |doi-access=free |article-number=e13471 |number=6}}</ref>

In mathematics, the notion of cardinality was first introduced by [[Georg Cantor]] in the late 19th century, wherein he used the used the term ''Mächtigkeit'', which may be translated as "magnitude" or "power", though Cantor credited the term to a work by [[Jakob Steiner]] on [[projective geometry]].<ref name=":2">{{Cite book |last=Ferreirós |first=José |url=https://link.springer.com/book/10.1007/978-3-7643-8350-3 |title=Labyrinth of Thought |date=2007 |publisher=[[Birkhäuser]] |isbn=978-3-7643-8349-7 |edition=2nd |location=Basel |pages=24 |doi=10.1007/978-3-7643-8350-3}}</ref><ref name=":3">{{Cite journal |last=Cantor |first=Georg |author-link=Georg Cantor |date=1932 |editor-last=Zermelo |editor-first=Ernst |editor-link=Ernst Zermelo |title=Gesammelte Abhandlungen |url=https://link.springer.com/book/10.1007/978-3-662-00274-2 |journal=Springer |location=Berlin |pages=151 |doi=10.1007/978-3-662-00274-2 |isbn=978-3-662-00254-4}}</ref><ref name=":4">{{Cite book |last=Steiner |first=Jacob |url=https://archive.org/details/bub_gb_jCgPAAAAQAAJ/ |title=Vorlesungen über synthetische Geometrie / 1 Die Theorie der Kegelschnitte in elementarer Form |date=1867 |publisher=Leipzig : Teubner |others=Ghent University}}</ref> The terms ''cardinality'' and ''cardinal number'' were eventually adopted from the grammatical sense, and later translations would use these terms.<ref>Harper Douglas, "Etymology of cardinality," [[Etymonline]], Online Etymology Dictionary, accessed April 20, 2025, https://www.etymonline.com/word/cardinality.</ref><ref>[[Oxford English Dictionary]], "cardinality (n.2), Etymology," March 2025, https://doi.org/10.1093/OED/5444748676.</ref> Similarly, the terms for ''countable'' and ''uncountable sets'' come from [[Countable noun|''countable'']] and ''[[uncountable nouns]]''.{{citation needed|date=April 2025|reason=parallels are obvious, but is the use of "countable" +/- "un" for sets derived from words used for nouns, or is it original research?}}