set theory - What's wrong with the surreals?

I just read the epilogue of On Numbers and Games as suggested by Timothy Chow, and here I see very concrete references to some technical difficulties that may demystify some research in this area, although it is apparent by other answers that there is still much optimism.

Some remarks based on the epilogue (written in 2000 by Conway):

There is a nice definition of the surreals which does not require equality as a defined relation. It is not formally given, but I'm guessing we can think of it as a mapping from an ordinal set to signs {-,+}, each sign being a direction we take in the surreal tree. Then identity is equality.

However, Conway remarks that this has two problems:

1. It forsakes the "genetic" (his word, also in quotes at first) approach of the L,R definition. I don't fully understand this, but I'm guessing he means that we're building everything on the intuition of a total ordering (and maybe a "time-of-creation" idea), and the surreals will always be identifiable with L,R sets, so why not just define them that way?




2. The sign-sequence definition requires that the ordinals are defined first.

Conway goes on to discuss work by Simon Norton (a proposed definition of an integral) and Martin Kruskal.

The general direction here is to define things in terms of (L,R) sets (classes??) in such a way that equal numbers (in the defined equality) give equal answers; and that classical analysis remains intact.

Conway gives Norton's integral definition, which has some good properties, but fails to integrate the surreal-exponential function in accordance with classical analysis (we get e^x instead of e^x-1 when integrating over [0,x]).

In summary, I'm choosing to interpret all of these comments and answers together (thanks to all) as: the surreals are indeed a worthwhile construction, although there is a noted lack of progress on extending calculus to work equally elegantly in a surreal-general setting.

In case others are curious, here are some references (I have read none of them, yet) Conway gives in this epilogue:

• The Theory of Surreal Numbers by Harry Gonshor


• Foundations of Analysis over Surreal Number Fields by Norman Alling


• Real Numbers, Generalizations of the Reals, and Theories of Continua by Philip Ehrlich