Editing Transfinite number (section)

{{Short description|Number that is larger than all finite numbers}}
In [[mathematics]], '''transfinite numbers''' or '''infinite numbers''' are numbers that are "[[Infinity|infinite]]" in the sense that they are larger than all [[finite set|finite]] numbers. These include the '''transfinite cardinals''', which are [[cardinal number]]s used to quantify the size of infinite sets, and the '''transfinite ordinals''', which are [[ordinal number]]s used to provide an ordering of infinite sets.<ref>{{Cite web|url=https://www.dictionary.com/browse/transfinite-number|title=Definition of transfinite number {{!}} Dictionary.com|website=www.dictionary.com|language=en|access-date=2019-12-04}}</ref><ref name=":0">{{Cite web|url=https://www.math.utah.edu/~pa/math/sets.html|title=Transfinite Numbers and Set Theory|website=www.math.utah.edu|access-date=2019-12-04}}</ref> The term ''transfinite'' was coined in 1895 by [[Georg Cantor]],<ref>{{Cite web|url=https://www.britannica.com/biography/Georg-Ferdinand-Ludwig-Philipp-Cantor|title=Georg Cantor {{!}} Biography, Contributions, Books, & Facts|website=Encyclopedia Britannica|language=en|access-date=2019-12-04}}</ref><ref>{{cite journal | url=http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN00225557X | author=Georg Cantor | title=Beiträge zur Begründung der transfiniten Mengenlehre (1) | journal=Mathematische Annalen | volume=46 | number=4 | pages=481&ndash;512 | date=Nov 1895 }} {{Open access}}</ref><ref>{{cite journal | url=http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002256460 | author=Georg Cantor | title=Beiträge zur Begründung der transfiniten Mengenlehre (2) | journal=Mathematische Annalen | volume=49 | number=2 | pages=207&ndash;246 | date=Jul 1897 }} {{Open access}}</ref><ref>{{cite book | url=https://www.maths.ed.ac.uk/~v1ranick/papers/cantor1.pdf | author=Georg Cantor | editor=Philip E.B. Jourdain | title=Contributions to the Founding of the Theory of Transfinite Numbers | location=New York | publisher=Dover Publications, Inc. | year=1915 }} English translation of Cantor (1895, 1897).</ref> who wished to avoid some of the implications of the word ''infinite'' in connection with these objects, which were, nevertheless, not ''finite''.{{cn|reason=Encyclopedia Britannica[4] doesn't cover this aspect. Cantor.1915[7] apparently just distinguishes 'finite aggregates' and 'transfinite aggregates' (p. 103, Sect. 6).|date=May 2021}} Few contemporary writers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as ''infinite numbers''. Nevertheless, the term ''transfinite'' also remains in use.

Notable work on transfinite numbers was done by [[Wacław Sierpiński]]: ''Leçons sur les nombres transfinis'' (1928 book) much expanded into ''[[Cardinal and Ordinal Numbers]]'' (1958,<ref name=oxtoby>{{citation

 | last = Oxtoby | first = J. C. | authorlink = John C. Oxtoby
 | doi = 10.1090/S0002-9904-1959-10264-0
 | issue = 1
 | journal = [[Bulletin of the American Mathematical Society]]
 | mr = 1565962
 | pages = 21–23
 | title = Review of ''Cardinal and Ordinal Numbers'' (1st ed.)
 | volume = 65
 | year = 1959| doi-access = free
 }}</ref> 2nd ed. 1965<ref name=goodstein>{{citation
 | last = Goodstein | first = R. L. | authorlink = Reuben Goodstein
 | date = December 1966
 | doi = 10.2307/3613997
 | issue = 374
 | journal = [[The Mathematical Gazette]]
 | jstor = 3613997
 | page = 437
 | title = Review of ''Cardinal and Ordinal Numbers'' (2nd ed.)
 | volume = 50}}</ref>).