FOM: examples of infinitesimals

Vladimir Sazonov sazonov at
Tue Nov 25 15:53:49 EST 1997

Stephen G Simpson wrote:
>   Several people have said that it's easy to give examples of
>   infinitesimals in nonstandard analysis.  But nobody seems to have
>   commented on Harvey's highly relevant posting
>      FOM: 7:Undefinability/Nonstandard Models
>   on November 17.  The essential point there is that it's impossible
>   to give an example of an infinitesimal under the Abraham Robinson
>   setup, because it's consistent with ZFC that there is no definable
>   nonprincipal ultrafilter on N, the natural numbers.  If e were a
>   definable infinitesimal, then
>      {X subset of N: [1/e] is an element of X*}
>   would be a definable nonprincipal ultrafilter on N.

Yes, it is clear. However, it seems that

	(i) either we should explicitly declare that when discussing 
on natural or real numbers, etc., we mean just the ordinal \omega 
from the theory ZFC or the like; then the question of non-uniqueness 
of hyperreals or undefinability of specific examples of 
infinitesimals becomes more technical than foundational one (or 
foundational *modulus* ZFC, which is, of course, also interesting),

	(ii) or we are discussing these notions from a *wider
perspective*; in this case essentially NO a priori uniqueness of 
natural numbers can be asserted (and even clearly articulated) and 
the question on definability of specific examples of infinitesimals
may crucially depend on some, possibly *alternative* to ZFC, approach 
which may be even non-reducible to (any extension of) ZFC in any 
direct way.

I think that Robinson's approach to Nonstandard Analysis (NA) is 
only one possibility to formalize infinitesimals. There may be 
others, rather different approaches which could give different 
answers to our "foundational" questions. As to concrete examples 
of infinitesimals, we are actually dealing with them in our 
everyday life. (Take a grain of sand on a beach or, say, the length 
10^{-40}cm - quite concrete.) 

Thus, we probably need corresponding theory of *such* 
infinitesimals if, by some reasons, we are not sufficiently 
satisfied with the Standard Analysis and in the ordinary version 
of NA (say, because of undefinability or non-uniqueness of 
infinitesimals). No doubts, the interest to any NA has some 
*real grounds*. It is this "nonstandard" way which we *understand* 
Analysis and its theorems on the base of our everyday geometrical 
and physical practice and intuition.  But traditional f.o.m. in the 
form of ZFC forces us to *prove* them somewhat differently, by using 
epsilon-delta.  Even formalization of NA in ZFC, which is, of course, 
a great achievement, proves to be not very satisfactory in the
abovementioned sense. 

Of course, from the point of view of the ordinary mathematics based 
on (or compatible with) ZFC the above examples of infinitesimals 
are considered as something "incorrect" or as "nonsense", etc. But 
we may, at least, *try* to find different foundations where this 
will be correct. I actually presented shortly some such attempt 
(I realize that it is rather primitive at the present state, 
however completely *rigorous*) in my previous postings to FOM on 
feasible numbers from 05 Nov 1997, 05 Nov 1997 and 12 Nov 1997.
Another approach mentioned in the posting of Rick Sommer
<sommer at Csli.Stanford.EDU> from 11 Nov 1997 seems also related. 

Vladimir Sazonov

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