[FOM] 622: Adventures in Formalization 6

[Freek Wiedijk]

Dear John,

>> I think the Conway numbers don't work well with
>> exponentiation?  You cannot take the omega-logarithm in a
>> way that omega-exponentiation gives you back your original
>> number?
>
>This is incorrect. Ehrlich and Van den Dries showed the
>exponential ordered field of surreal numbers is an
>elementary extension of the exponential ordered field of
>real numbers.

My remark was based on the sentence

  We do not know of any natural power operation x^y defined
  for all numbers y and all postive x, [...]

in the section named "the omega-map" on page 31 of my copy
of On Numbers and Games.  And then, at the very end of
chapter 3:

  Note added in second printing:
  [...] and this does yield a definiton of analytic powers
  x^y, via the natural logarithm defined by integrating 1/x.
  The function omega^x of the text is however an _ordinal_
  power, not this analytic one.

So omega^x in one sense is not omega^x in the other sense,
even though both are defined voor all surreal numbers x?
That's why I thought: "exponentiation of surreal numbers
is not very natural".

Also, this integral of 1/x definition seems very different
from the definitions of addition and multiplication on the
surreal numbers, which is not very attractive either.

Freek