Numbers...

I've been on this list for a while... but only followed it passively
as a lurker, and even then not carefully all the time. Ok, that's my
wonderful introduction I guess.

Anyway, I like this thing below, and wonder if anyone knows more:

> John Conway, the nutty mathematician who created the Game of Life, a
> primal ancestor to today's artifical life programs, got his inspiration
> from Go. Studying GO games, Conway realized that at the end of the game
> "the whole game looked like a sum of many little games" [Sciences
> May/June 1994]. From analyzing the mathematical properties of Go and
> other number-games, Conway came up with the set of "surreal numbers,"
> which I cannot even vaguely understand, though the "explanation" goes
> like this: "what real numbers are to integars, surreal numbers are to
> transfinite numbers." Oh.

I take it that the relation of real numbers to integers is that real
numbers are (named by) equivalence classes of infinite series of
ordered pairs of integers.

That is, ordered pairs of integers -- i.e. ratios -- define the
rational numbers, like 3/4, 77/3, etc. Sometimes infinite series of
these rational numbers *converge* to numbers which are not themselves
rational. These are called Cauchy sequences. The way to prove this
is to order the rational numbers in an integer-like order, i.e. by
Cantor diagnalization. Then start a Cauchy sequence, and for each
number in the ordered sequence of rationals, add to the Cauchy
sequence in a manner such as to "keep away from" the next rational.
I..e. suppose I start with the (finite) sequence, {1/2, 3/4, 7/8}, and
that according to my ordering of the rationals, the next rational I'm
trying to *avoid* is 15/16. I can assure that my Cauchy sequence does
NOT converge to 15/16 by making sure that every additional element
added to the Cauchy sequence is less than 1/(16+1) away from 7/8. By
doing this iteratively for the enumerated rationals, we can construct
a sequence not converging to any rational number. We have created an
*irrational* number. Of course, the reals include both the rationals
and the irrationals... but the one's that are "special" are the
irrationals, since we already knew there were rational numbers.

My *guess* is that surreal numbers are also sequences of transfinite
numbers. That would be the obvious analogy, IMO. (I don't think the
intermediary step about ordered pairs is probably in there).
Probably, Conway took (transfinite) sequences of transfinite
cardinals, and somehow *excluded* or *avoided* those sequences which
converged to other transfinite cardinals. Although I can't think how
to prove it, it seems like there would be such sequences -- I vaguely
remember a discussion of such a thing in my transfinite set-theory
class of many years ago. My guess is that the surreal numbers are
these "cardinal-avoiding" sequences of cardinals. Of course, you
can't avoid ordinals, so I'm sure that's not the issue. Or maybe the
surreals include the cardinals, like the reals include the rationals.
I dunno.

Anyone know whether my guess is along the right line? Anyone
understand my guess above?

Yours, Lulu...

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