Editing Surreal number (section)

===Simplicity hierarchy===

A construction of the surreal numbers as a maximal binary pseudo-tree with simplicity (ancestor) and ordering relations is due to Philip Ehrlich.<ref name="Ehr12">{{cite journal
 |author      = Philip Ehrlich
 |author-link  = Philip Ehrlich
 |year        = 2012
 |title       = The absolute arithmetic continuum and the unification of all numbers great and small
 |journal     = The Bulletin of Symbolic Logic
 |volume      = 18
 |issue       = 1
 |pages       = 1–45
 |url         = http://www.ohio.edu/people/ehrlich/Unification.pdf
 |access-date  = 2017-06-08
 |doi         = 10.2178/bsl/1327328438
 |s2cid = 18683932
 |url-status     = dead
 |archive-url  = https://web.archive.org/web/20171007095144/http://www.ohio.edu/people/ehrlich/Unification.pdf
 |archive-date = 2017-10-07
}}</ref> The difference from the usual definition of a tree is that the set of ancestors of a vertex is [[well-order]]ed, but may not have a [[maximal element]] (immediate predecessor); in other words the order type of that set is a general ordinal number, not just a natural number. This construction fulfills Alling's axioms as well and can easily be mapped to the sign-sequence representation. Ehrlich additionally constructed an isomorphism between Conway's maximal surreal number field and the maximal [[hyperreal field|hyperreals]] in [[von Neumann–Bernays–Gödel set theory]].<ref name="Ehr12" />