FOM: Surreal structure
Joe Shipman
shipman at savera.com
Wed May 26 14:15:15 EDT 1999
Simpson:
>The techniques of nonstandard analysis are based on model-theoretic
>constructions, specifically saturated real closed fields with
>additional structure. (The term ``hyperreals'' is used.) By my FOM
>posting of 21 May 1999 19:59:44, saturated real closed fields are the
>same thing as surreal numbers.
>
>Does anybody know whether either of these two groups (Conway et al,
>Keisler et al) has ever acknowledged the existence of the other?
Steve, the previous post of Holmes's answers this, and I agree with
Holmes's other comments as well (especially "This is a wonderful book,
and my interest in the foundations of mathematics was strongly enhanced
by reading it as an undergraduate.", which I could have said myself). I
would just like to reiterate that "saturated real closed fields" are
only "the same thing as surreal numbers" if you ignore the most
interesting thing about the surreal numbers, which is the explicit
construction {A|B} of a new surreal number from two sets of surreal
numbers satisfying an ordering condition.
Furthermore, if you move on to Part One of the book (the "Games" part,
Part Zero is the "Numbers" part) you find that this hierarchical
inductive construction is not simply an alternative way to get a
structure, the elements of which are thenceforth treated as if they were
all given at once -- it is absolutely of the essence because of the
natural correspondence of the left and right "members" of a game with
the "moves" or "options" of the game. Conway's theory of Games only
makes sense in light of this added structure, although for Numbers it is
possible to restrict one's attention to the properties No inherits as a
saturated real-closed Field and not revisit the original construction.
Two other points of foundational interest:
1) The Structure of "impartial games" (games which treat the players
symmetrically so the same moves are available to both) is isomorphic to
the set-theoretic Universe V. A game is the set of its options and a
"move" is choosing an element. There is an amazingly interesting
function from sets to ordinals, given by taking the "nim-value" of the
game (value of the function on a set is the least ordinal not the value
of the function on any element of the set). This function is like the
ordinal rank function except that it takes the least excluded value
rather than the sup.
2) The game-theoretic operations of nim-addition and nim-multiplication
can be defined for all ordinals by a formula. The initial segments of
the resulting structure are also amazingly interesting (see chapter 6),
giving very explicit constructions of certain structures of
characteristic 2; see my reply to Forster from May 4th, part of which I
reproduce here:
A better example of double exponential growth comes from Conway: the
finite ordinals which form a field under nim-addition and
nim-multiplication are those of the form 2^(2^n) for n=0,1,2,3,... (2,
4, 16, 256, 65536, etc., where each term is the square of the previous
one).
.....
The infinite ordinals which are fields under the Nim operations are much
more interesting. omega^(omega^omega) is the first algebraically closed
field under the Nim-operations (that is, the ordinal omega^(omega^omega)
is the first ordinal transcendental over the earlier ones); the next
transcendental is very large, and Conway leaves as an open question what
its relationship is to the first impredicative ordinal Gamma_0.
-- Joe Shipman
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