Editing Hyperreal number (section)

== Properties ==
The hyperreals *'''R''' form an [[ordered field]] containing the reals '''R''' as a [[Field extension|subfield]]. Unlike the reals, the hyperreals do not form a standard [[metric space]], but by virtue of their order they carry an [[order topology]].

The use of the definite article ''the'' in the phrase ''the hyperreal numbers'' is somewhat misleading in that there is not a unique ordered field that is referred to in most treatments. However, a 2003 paper by [[Vladimir Kanovei]] and [[Saharon Shelah]]<ref name=kanovei2003>{{Citation| last1=Kanovei| first1=Vladimir| last2=Shelah| first2=Saharon| title=A definable nonstandard model of the reals| url=http://shelah.logic.at/files/825.pdf| journal=Journal of Symbolic Logic| volume=69| year=2004| pages=159–164| doi=10.2178/jsl/1080938834| arxiv=math/0311165| s2cid=15104702| access-date=2004-10-13| archive-url=https://web.archive.org/web/20040805172214/http://shelah.logic.at/files/825.pdf| archive-date=2004-08-05| url-status=dead}}</ref> shows that there is a definable, countably [[saturated model|saturated]] (meaning [[ω-saturated]] but not [[Countable set|countable]]) [[elementary substructure|elementary extension]] of the reals, which therefore has a good claim to the title of ''the'' hyperreal numbers.  Furthermore, the field obtained by the ultrapower construction from the space of all real sequences, is unique up to isomorphism if one assumes the [[continuum hypothesis]].

The condition of being a hyperreal field is a stronger one than that of being a [[real closed field]] strictly containing '''R'''. It is also stronger than that of being a [[superreal field]] in the sense of Dales and [[W. Hugh Woodin|Woodin]].<ref>{{Citation | last1=Woodin | first1=W. H. | last2=Dales | first2=H. G. | title=Super-real fields: totally ordered fields with additional structure | publisher=Clarendon Press | location=Oxford | isbn=978-0-19-853991-9 | year=1996}}</ref>