Editing Surreal number (section)

== History of the concept ==
Research on the [[Go strategy and tactics|Go endgame]] by [[John Horton Conway]] led to the original definition and construction of the surreal numbers.<ref name="O'Connor">{{citation | url = http://www-history.mcs.st-andrews.ac.uk/Biographies/Conway.html | title = John Horton Conway | last1 = O'Connor | first1 = J.J. | last2 = Robertson | first2 = E.F. | access-date = 2008-01-24 | date= June 2004 | archive-url =http://web.archive.org/web/20080314152337/http://www-history.mcs.st-andrews.ac.uk/Biographies/Conway.html | archive-date = 14 March 2008 | work= School of Mathematics and Statistics | publisher= University of St Andrews, Scotland | url-status= dead}}</ref>  Conway's construction was introduced in [[Donald Knuth]]'s 1974 book ''Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness''. In his book, which takes the form of a dialogue, Knuth coined the term ''surreal numbers'' for what Conway had called simply ''numbers''.<ref>{{cite web |last1=Knuth |first1=Donald |title=Surreal Numbers |url=https://www-cs-faculty.stanford.edu/~knuth/sn.html |publisher=Stanford |access-date=25 May 2020}}</ref>  Conway later adopted Knuth's term, and used surreals for analyzing games in his 1976 book ''[[On Numbers and Games]]''.

A separate route to defining the surreals began in 1907, when [[Hans Hahn (mathematician)|Hans Hahn]] introduced [[Hahn series]] as a generalization of [[formal power series]], and [[Felix Hausdorff]] introduced certain ordered sets called [[η set|{{math|''η''{{sub|''α''}}}}-sets]] for ordinals {{mvar|α}} and asked if it was possible to find a compatible ordered group or field structure. In 1962, Norman Alling used a modified form of Hahn series to construct such ordered fields associated to certain ordinals {{mvar|α}} and, in 1987, he showed that taking {{mvar|α}} to be the class of all ordinals in his construction gives a class that is an ordered field isomorphic to the surreal numbers.<ref>{{citation|title=On the existence of real-closed fields that are {{math|''η''{{sub|''α''}}}}-sets of power&nbsp;{{math|ℵ{{sub|''α''}}}}.
|first= Norman L.|last= Alling 
|journal= Trans. Amer. Math. Soc.|volume= 103 |year=1962|pages= 341–352 
|mr= 0146089|doi=10.1090/S0002-9947-1962-0146089-X|doi-access= free}}</ref>

If the surreals are considered as 'just' a proper-class-sized real closed field, Alling's 1962 paper handles the case of [[Inaccessible cardinal|strongly inaccessible]] cardinals which can naturally be considered as proper classes by cutting off the [[Von Neumann universe|cumulative hierarchy of the universe]] one stage above the cardinal, and Alling accordingly deserves much credit for the discovery/invention of the surreals in this sense. There is an important additional field structure on the surreals that isn't visible through this lens however, namely the notion of a 'birthday' and the corresponding natural description of the surreals as the result of a cut-filling process along their birthdays given by Conway. This additional structure has become fundamental to a modern understanding of the surreal numbers, and Conway is thus given credit for discovering the surreals as we know them today—Alling himself gives Conway full credit in a 1985 paper preceding his book on the subject.<ref name="Alling1985">{{citation | url = https://www.ams.org/journals/tran/1985-287-01/S0002-9947-1985-0766225-7/S0002-9947-1985-0766225-7.pdf | title = Conway's Field of surreal numbers | last = Alling | first = Norman | date = Jan 1985 | journal = Trans. Amer. Math. Soc. | volume = 287 | issue = 1 | pages = 365–386 | access-date = 2019-03-05 | doi=10.1090/s0002-9947-1985-0766225-7| doi-access = free }}</ref>