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{{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=481–512 | 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=207–246 | 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 particular he believed that "truly infinite" is a perfect and thus divine quality and so refused to attribute this term to mathematical constructs comprehensible by humans.<ref>{{Cite journal |last=Tapp |first=Christian |date=2005-08-01 |title=On Some Philosophical Aspects of the Background to Georg Cantor’s theory of sets |url=https://journals.openedition.org/philosophiascientiae/386?lang=en |journal=Philosophia Scientiæ. Travaux d'histoire et de philosophie des sciences |language=en |issue=CS 5 |pages=157–173 |doi=10.4000/philosophiascientiae.386 |issn=1281-2463}}</ref> 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>{{cite journal

| 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>{{cite journal
| 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>).

==Definition==
Any finite [[natural number]] can be used in at least two ways: as an ordinal and as a cardinal. Cardinal numbers specify the size of sets (e.g., a bag of {{Em|five}} marbles), whereas ordinal numbers specify the order of a member within an ordered set<ref name=":1">{{Cite web|url=http://mathworld.wolfram.com/OrdinalNumber.html|title=Ordinal Number|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|date=3 May 2023}}</ref> (e.g., "the {{Em|third}} man from the left" or "the {{Em|twenty-seventh}} day of January"). When extended to transfinite numbers, these two concepts are no longer in [[one-to-one correspondence]]. A transfinite cardinal number is used to describe the size of an infinitely large set,<ref name=":0" /> while a transfinite ordinal is used to describe the location within an infinitely large set that is ordered.<ref name=":1" />{{not in citation given|date=May 2021}} The most notable ordinal and cardinal numbers are, respectively:

*<math>\omega</math> ([[Ordinal number#Ordinals extend the natural numbers|Omega]]): the lowest transfinite ordinal number. It is also the [[order type]] of the [[natural number]]s under their usual linear ordering.
*<math>\aleph_0 </math> ([[Aleph-null]]): the first transfinite cardinal number. It is also the [[cardinality]] of the natural numbers. If the [[axiom of choice]] holds, the next higher cardinal number is [[aleph-one]], <math>\aleph_1.</math> If not, there may be other cardinals which are incomparable with aleph-one and larger than aleph-null. Either way, there are no cardinals between aleph-null and aleph-one.

The [[continuum hypothesis]] is the proposition that there are no intermediate cardinal numbers between <math>\aleph_0</math> and the [[cardinality of the continuum]] (the cardinality of the set of [[real number]]s):<ref name=":0" /> or equivalently that <math>\aleph_1</math> is the cardinality of the set of real numbers. In [[Zermelo–Fraenkel set theory]], neither the continuum hypothesis nor its negation can be proved.

Some authors, including P. Suppes and J. Rubin, use the term ''transfinite cardinal'' to refer to the cardinality of a [[Dedekind-infinite set]] in contexts where this may not be equivalent to "infinite cardinal"; that is, in contexts where the [[axiom of countable choice]] is not assumed or is not known to hold. Given this definition, the following are all equivalent:
* <math>\mathfrak{m}</math> is a transfinite cardinal. That is, there is a Dedekind infinite set <math>A</math> such that the cardinality of ''<math>A</math>'' is <math>\mathfrak {m}.</math>
* <math>\mathfrak{m} + 1 = \mathfrak{m}.</math>
* <math>\aleph_0 \leq \mathfrak{m}.</math>
* There is a cardinal <math>\mathfrak{n}</math> such that <math>\aleph_0 + \mathfrak{n} = \mathfrak{m}.</math>

Although transfinite ordinals and cardinals both generalize only the natural numbers, other systems of numbers, including the [[hyperreal number]]s and [[surreal number]]s, provide generalizations of the [[real number]]s.<ref>{{cite journal
| last1 = Beyer | first1 = W. A.
| last2 = Louck | first2 = J. D.
| doi = 10.1006/aama.1996.0513
| issue = 3
| journal = Advances in Applied Mathematics
| mr = 1436485
| pages = 333–350
| title = Transfinite function iteration and surreal numbers
| volume = 18
| year = 1997| doi-access = free
}}</ref>

==Examples==
In Cantor's theory of ordinal numbers, every integer number must have a successor.<ref name="ONG">[[John Horton Conway]], (1976) ''[[On Numbers and Games]]''. Academic Press, ISBN 0-12-186350-6. ''(See Chapter 3.)''</ref> The next integer after all the regular ones, that is the first infinite integer, is named <math>\omega</math>. In this context, <math>\omega+1</math> is larger than <math>\omega</math>, and <math>\omega\cdot2</math>, <math>\omega^{2}</math> and <math>\omega^{\omega}</math> are larger still. Arithmetic expressions containing <math>\omega</math> specify an ordinal number, and can be thought of as the set of all integers up to that number. A given number generally has multiple expressions that represent it, however, there is a unique [[Ordinal arithmetic#Cantor normal form|Cantor normal form]] that represents it,<ref name="ONG" /> essentially a finite sequence of digits that give coefficients of descending powers of <math>\omega</math>.

Not all infinite integers can be represented by a Cantor normal form however, and the first one that cannot is given by the limit <math>\omega^{\omega^{\omega^{...}}}</math> and is termed <math>\varepsilon_{0}</math>.<ref name="ONG" /> <math>\varepsilon_{0}</math> is the smallest solution to <math>\omega^{\varepsilon}=\varepsilon</math>, and the following solutions <math>\varepsilon_{1}, ...,\varepsilon_{\omega}, ...,\varepsilon_{\varepsilon_{0}}, ...</math> give larger ordinals still, and can be followed until one reaches the limit <math>\varepsilon_{\varepsilon_{\varepsilon_{...}}}</math>, which is the first solution to <math>\varepsilon_{\alpha}=\alpha</math>. This means that in order to be able to specify all transfinite integers, one must think up an infinite sequence of names: because if one were to specify a single largest integer, one would then always be able to mention its larger successor. But as noted by Cantor,{{citation needed|date=May 2021}} even this only allows one to reach the lowest class of transfinite numbers: those whose size of sets correspond to the cardinal number <math>\aleph_{0}</math>.

==See also==
{{Wiktionary|transfinite}}
{{div col|colwidth=20em}}
*[[Actual infinity]]
*[[Aleph number]]
*[[Beth number]]
*[[Epsilon number]]
*[[Infinitesimal]]
*[[Transfinite induction]]
{{div col end}}

== References ==
{{Reflist}}

==Bibliography==

*Levy, Azriel, 2002 (1978) ''Basic Set Theory''. Dover Publications. {{isbn|0-486-42079-5}}
*O'Connor, J. J. and E. F. Robertson (1998) "[http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Cantor.html Georg Ferdinand Ludwig Philipp Cantor]," [[MacTutor History of Mathematics archive]].
*[[Jean E. Rubin|Rubin, Jean E.]], 1967. "Set Theory for the Mathematician". San Francisco: Holden-Day. Grounded in [[Morse–Kelley set theory]].
*[[Rudy Rucker]], 2005 (1982) ''Infinity and the Mind''. Princeton Univ. Press. Primarily an exploration of the philosophical implications of [[Cantor's paradise]]. {{isbn|978-0-691-00172-2}}.
*[[Patrick Suppes]], 1972 (1960) "[https://books.google.com/books?id=sxr4LrgJGeAC Axiomatic Set Theory]". Dover. {{isbn|0-486-61630-4}}. Grounded in [[ZFC]].

{{Large numbers}}
{{Infinity}}
{{Authority control}}

[[Category:Basic concepts in infinite set theory]]
[[Category:Cardinal numbers]]
[[Category:Ordinal numbers]]

Transfinite number - Revision history

imported>Vonfraginoff: fix typo