From: karr@cs.cornell.edu (David Karr)
Subject: Re: more on .999... = 10.0
Organization: Cornell Univ. CS Dept, Ithaca NY 14853
Distribution: inet
Date: Wed, 30 Sep 1992 01:37:20 GMT

In article <1a9qi2INN898@uwm.edu> markh@csd4.csd.uwm.edu (Mark) writes:
>[David Karr wrote:]
>>What twentieth-century models of the real numbers have [infinitesimals]?
>
>You must have been asleep for 30 years.  Producing such a model is trivial.

31 years to be more precise.  Someone was just kind enough to lend me
an account of a real number system with infinitesimals, based on work
by Abraham Robinson in 1961 (in Enderton, A Mathematical Introduction
to Logic).  It's actually a very interesting account.  Particularly
interesting is the fact you can prove theorems identical to theorems
of integral calculus in this system.

What still bothers me, however, is what is the meaning of .999... in
*R (reals with infinitesimals).  I'd thought that .999... was a
representation for the sup of the set

  A = { .9, .99, .999, .9999, ... }

That is, the set of all 9/10 + 9/100 + ... + 9/10^n for n in N.  But
in a system of real numbers with infinitesimals, for every element z
of A, z *< 1 - i for any infinitesimal i, yet there is some z > 1 - r
for any "standard" number r.  So there would appear to be no supremum
of A.  In fact Enderton warns that *R does not have the
least-upper-bound property, i.e. there will be subsets of *R with no
supremum.

Without being able to take a supremum, I do not know how to say that
ANY non-terminating decimal (other than one terminating in all 1's)
is equal to a real number (as opposed to just infinitely close).
That means that any number not equal to an integer divided by a
power of ten has no exact decimal representation.

To be honest, I don't view that as a fatal flaw in the number system.
All it does is throw out some notions about non-terminating decimals.
Perhaps it requires some rethinking of the famous diagonalization
theorem showing the reals are uncountable.  I don't know any reason
why we need decimal representations of every number to do math.

However, if you can actually come up with a convincing description of
how to interpret .999... correctly with infinitesimals, I'd be
interested.  (I'm not convinced by the interpretation sum_k=1^epsilon
9/10^k where epsilon is infinite.  In the "standard" interpretation
the set A had partial sums fo the form sum_k=1^n where n in N,
therefore n is finite.  Even if you replace N with *N (i.e. naturals
with infinites) then if n in *N, also n+1 in *N, so
sum_k=1^(epsilon+1) 9/10^k is also in A and the sum up to epsilon is
not an upper bound.  In plain language, once you've agreed to an
unending series of 9's, there's no justification for ending it, even
at an infinite number.)

>>>So choose any garden variety interpretation of the real number system, it has
>>>infinitesimals in it with probability 1.

I'm not sure I would qualify Robinson's work as "garden variety."

>>If you take a random selection of models of real numbers out of the
>>many analysis books on my shelf, I believe you will find
>>infinitesimals in it with probability 0.
>
>Wrong.  A model is an infinite object, a book is finite.

I take it then you would not be able to show me one of your models
of reals with infinitesimals ...

More precisely, perhaps I should have written "descriptions of the
real numbers" instead of "models."  I pulled a sample off my shelf
(Apostol, Mathematical Analysis) to check.  Indeed Apostol writes
specifically that all elements of R are finite.  That implies none
except 0 is infinitesimal, for if i is infinitesimal and i != 0 then
1/i is infinite.  Apostol does extend R to R*, which includes infinite
numbers but only two of them: +infinity and -infinity.

Incidentally, Apostol's definition of infinite decimal representations
of real numbers mentions that
	1/8 = 0.125000...
and
	1/8 = 0.124999...
with the choice of which one to use depending on whether you have a
< or a <= at a certain point in your rule for generating the digits.
From this I derive 
	9.999... = 9.875 + 1/8 = 10 (exactly)
Of course this is an ipse dixit argument, so you can ignore it if
you like.

-- David Karr (karr@cs.cornell.edu)