[FOM] 622: Adventures in Formalization 6

John Baldwin jbaldwin at uic.edu
Fri Oct 30 18:18:35 EDT 2015

With apologies, I am going to include all of two previous messages as I
the context is needed.

On Fri, Oct 30, 2015 at 5:16 AM, Freek Wiedijk <freek at cs.ru.nl> wrote:

> 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

Phil Ehrlich wrote me the following:

Martin Kruskal introduced the natural exponentiation for the surreals and
Gonshor has an informative chapter devoted to it in his book on surreal
numbers. Conway originally doubted there is a natural exponentiation for
the surreals; however, by the second printing of the first edition of his
book he was forced to revise his view.

end quote:

Of course, there can be debate about `naturality' but I think that the
naturality of the analytically defined exponentiation
is witnessed by the Ehrlich Van den Dries proof that the surreal
exponential field is an elementary extension of the real exponential field.

Again, here is the reference: Fundamenta Mathematica 2001 Fields of Surreal
numbers and exponentiation

John T. Baldwin
Professor Emeritus
Department of Mathematics, Statistics,
and Computer Science M/C 249
jbaldwin at uic.edu
851 S. Morgan
Chicago IL

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