FOM: Canonicalness of the hyperreals, V, and V_alpha

[JSHIPMAN@bloomberg.net]

My favorite construction of the hyperreals is John H. Conway's in his classic
book "On Numbers and Games".  He marries the Cantorian construction of the
hierarchy of sets with Dedekind's method for constructing the reals as "cuts"
to create a beautiful and unique complete ordered Field (capitalized because
the domain is a proper Class) that will serve for doing nonstandard analysis
rigorously (dubbed "surreal numbers" by Knuth).  In one sense this is perfectly
canonical, but in another it is very far from canonical because the whole
universe of sets is dragged in.  The  natural fix is just to stop Conway's
construction at some ordinal stage--for  each suitable ordinal alpha there will
be a unique model of the hyperreals that is just as canonical as V_alpha is.
Conway's construction is like a set theory with two sorts of membership ("Left"
and "Right")--his general object is a "game" which has Left and Right members
that are games defined at earlier stages of the cumulative construction.
"Numbers" are equivalence classes of games for which an inductively defined "<="
makes sense; they naturally correspond to maps from ordinals to {L,R}.-J Shipman