From: "Ian Sutherland" <p00117@psilink.com>
Subject: Re: more on .999... = 10.0
Organization: Key Software, Inc.
Date: Wed, 30 Sep 1992 02:10:15 GMT
X-Mailer: PSILink (3.2)

The reader is advised to look at the attributions carefully; much of 
what I quote from David Karr's article was in turn quoted by him from 
"Mark's" article.

David Karr writes:
>In article <1a6tc7INNo0u@uwm.edu> markh@csd4.csd.uwm.edu (Mark) writes:
>
>>So ACTUALLY, x differs from 10 by an infinitesimal power of 10 as was pointed
>>out above.  Which decimal position it differs in depends on which 9.999...
>>you chose, since there's a huge number of distinct 9.999...'s that differ at
>>some infinite decimal position but are the same at all finite decimal positions.
>
>I am tempted to write, "Now we have heard it all."

Indeed.  It is quite true that in a nonstandard model of the FIRST ORDER 
theory of the reals, there are infinitely many numbers of the form 

	9.999...[nonstandardly many 9s]...9[all zeros after this]

None of these numbers are "nine point nine REPEATING".  There is only 
one 9.999... where the "..." means "a 9 in every place".  That unique 
number is equal to 10.0, in nonstandard models as well as standard ones.

>In article <1a6tjqINNo19@uwm.edu>, markh@csd4.csd.uwm.edu (Mark) goes on:
>
>>All models of the real number axioms have infinitesimals in them, except one.

All models of the FIRST ORDER real number axioms have infinitesimals in 
them except one.  What a mathematician means when he says "the reals" is 
not "some model of the first order real number axioms", he means "the 
unique COMPLETE model of the first order real number axioms".  There's 
only one of those, it's the only one that's complete, and it's the 
intended model for decimal representations.  It's only one you need to 
consider in discussing whether "9.9..." is equal to "10,0" or not.

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

This is an odd statement.  "Probability 1" in what measure space?  Even 
if you made sense out of this statement, what relevance does it have to 
expressions whose intended interpretation is over the one model which 
DOESN'T have infinitesimals?

>I know infinitesimals were utilized to prove many important theorems
>in the eighteenth century and thereabouts, but they also led people to
>"prove" many falsehoods.  I thought they were abandoned for that
>reason.  What twentieth-century models of the real numbers have them?

The use of infinitesimals in the eighteenth century was indeed 
inconsistent.  The idea of using infinitesimals was revived by Abraham 
Robinson and others in the 1960's (I think), leading to a kind of 
analysis called "nonstandard analysis".  In nonstandard analysis, you 
use nonstandard models which contain the standard reals to prove things 
about the standard reals.  It doesn't change anything about the standard 
reals though.  You can prove in nonstandard analysis that 9.99... = 
10.0, just as in standard analysis.

"Mark" responds to Mr. Karr:
>In article <1992Sep29.035048.4747@cs.cornell.edu> karr@cs.cornell.edu (David Karr) writes:
>>I am tempted to write, "Now we have heard it all."
>
>No you haven't.  The construction above is perfectly consistent, and the view
>is perfectly sound.

Unfortunately, it doesn't seem to be a correct analysis of the question 
"Is 9.99... = 10.0?" (or I misunderstand your argument).

>For that matter, it's an open question whether there even EXISTS a model of
>the real number system without infinitesimals in it.  As far as anyone knows,
>they all have infinitesimals in them.

If you adopt the definition of "infinitesimal" suggested by your 
construction of a nonstandard model (i.e., x is infinitesimal if it is 
the reciprocal of a number which is bigger than 1,2,3,...), then the 
above statement is incorrect.

>You must have been asleep for 30 years.  Producing such a model is trivial.

I think this is a trifle harsh, don't you?  Nonstandard models of the 
theory of the reals weren't exactly front page news, however trivial 
they may be to construct.

Xref: sparky sci.math:12278 rec.puzzles:6473