[FOM] 622: Adventures in Formalization 6
Freek Wiedijk
freek at cs.ru.nl
Fri Oct 30 06:16:23 EDT 2015
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
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