Editing Transfinite number (section)

==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>{{citation
 | 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>