FOM: Surreal Numbers

Joe Shipman shipman at
Fri May 21 13:20:09 EDT 1999

>I glanced through ``On Numbers and
>Games'' some more, but I didn't find where Conway states or proves a
>completeness property of the surreal numbers.  Could someone please
>help me with this?

This "property" is actually what defines the surreal numbers.  The
"Construction" on page 4, which comes before any theorems, is simply

If L,R are any two sets of numbers, and no member of L is >= any member
of R, then there is a number { L | R }.  All numbers are constructed in
this way.

The completeness "property" is that there are no gaps -- if the two sets
satisfy the ordering condition, there is a number in between them
(furthermore, a unique "simplest" such number).  The statement "All
numbers are constructed in this way" is a version of the Axiom of
Foundation for numbers.  Note that though the construction involves
sets, the set theory is naive and unspecified (in the later development
Conway gets more explicit about his set-theoretical axioms) and the
numbers {L | R} are not themselves "sets", although each number is
associated with two sets of numbers, its "left and right options".  It
is the insistence on a further degree of formalization so that the
numbers themselves actually ARE sets in a pure set theory with only the
epsilon symbol which Conway finds unnecessary and unenlightening.

>In my opinion, the most relevant criterion for evaluating FOM postings
>is not the quantity of heat, but rather something like the ratio of
>light to heat.

>heat = emotional pressure

>light = scientific enlightenment

I agree with this, but I think Felscher agrees with it also and was in
fact expressing an opinion on the ratio rather than the quantity of
heat.  I would criticize the collection of your posts on this topic
somewhat differently, emphasizing the ratio of reiteration to

-- Joe Shipman

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