[FOM] Nonstandard Methods

[Todd Wilson]

Replies to Friedman (30 Jul 2003 23:17:46 -0400, Nonstandard Methods)
and to Urquhart (31 Jul 2003 13:52:32 -0400, Smooth Infinitesimal
Analysis).

On Wed, 30 Jul 2003, Harvey Friedman wrote:
> It is now clear that there are rival theories of infinitesimal
> analysis, and there is not going to be any much of an agreement
> about what approach is most illuminating or most real or most
> appropriate.
> 
> I have a radical idea about what to do about such an impasse, and
> that is to get much more radical than everything else, in a setting
> where there cannot be any funny business going on.
> 
> I.e., do FINITE ANALYSIS ONLY. In fact, do FINITE MATHEMATICS ONLY.

In this connection, I would like to mention an old paper by sometime
FOM contributor Jan Mycielski:

    J. Mycielski, "Analysis without actual infinity", JSL 46:3, Sept
    1981, pp. 625-633.

Mycielski presents a formal system FIN that is locally finite (every
finite fragment has finite models) and is "sufficient for the
development of analysis in the same sense that ZFC ... is sufficient
for the development of mathematics".  The system includes a
rationally-indexed set of constants that represent (potentially)
infinite indiscernables in the theory, the reciprocals of which
function as infinitesimals, making the development of analysis in FIN
much like that in nonstandard analysis.
       
On Thu, 31 Jul 2003, Alasdair Urquhart wrote:
> Harvey Friedman raised the question of an axiomatic formulation of
> Smooth Infinitesimal Analysis.  Chapter VII of the monograph by
> Moerdijk and Reyes contains an axiomatic version of smooth analysis.

In addition to Moerdijk and Reyes, which remains a good reference, let
me mention that Bell's book, A Primer of Infinitesimal Analysis, also
devotes a chapter to presenting an axiomatic system for SIA, which,
for readers without immediate access to Bell's book, I reproduce here.
The formulation is given in the language of topos theory with a type
constant R, representing the smooth real line.  The logical axioms and
rules of inference are those of basic "topos logic" (an impredicative
intuitionistic type theory).  The other axioms are:

R1.  R has elements 0 and 1, unary operation -, and binary operations
     + and * that make R into a nontrivial field.

R2.  There is a relation < on R that makes it an ordered field in
     which square roots of positive elements can be extracted.

Define D = { x in R : x^2 = 0 }, the set of nilsquare infinitesimals.

SIA1.  For all f:D->R, there exists unique a,b in R such that for all
       e in D, f(e) = a + be.

This is the Principle of Microaffiness, which says that all functions
are linear at infinitesimal scale.

SIA2.  For all f:R->R, if f satisfies f(x+e)=f(x) for all x in R and e
       in D, then f satisfies f(x) = f(y) for all x,y in R.

This is the Constancy Principle, which says that a function that is
infinitesimally constant is everywhere constant.  (Note that these
axioms are inconsistent with the Law of Excluded Middle, since
functions violating them can be defined by cases involving undecidable
predicates.)  Additional axioms can be added to facilitate integration
and higher-order (as opposed to only nilsquare) infinitesimals:

IP.  For all f:[0,1]->R, there is a unique g:[0,1]->R such that g' = f
     and g(0)=0.

Here, the derivative g' of g is defined to be the unique function that
satisfies

    g(x+e) = g(x) + e g'(x)

for all x in R and e in D, guaranteed to exist by SIA1.  IP is the
Integration Principle.  Define D_k = { x in R : x^k = 0 }, the set of
kth-order infinitesimals.

MP.  For any k >= 1 and g:D_k -> R, there exists unique b_1, ..., b_k
     in R such that for all d in D_k, we have

     g(d) = g(0) + Sum_{n=1}^k (b_n d^n)

This is the Principle of Micropolynomiality, which generalizes
Microaffiness to higher order infinitesimals and allows a version of
Taylor's Theorem to be developed involving iterated derivatives.

The set N of natural numbers can be defined within SIA as the smallest
set containing 0 and closed under adding 1, which therefore satisfies
the full induction principle.  However, in models where R is neither
Archimedean nor compact, it is often more natural to consider the
*smooth* natural numbers:

    N* = { x in R : x >= 0 and sin(pi*x) = 0 }.

Then, N* enjoys many of the properties of N, and R is always
Archimedean and compact with respect to N*, i.e., when N* is used in
the definitions of these concepts instead of N.

Finally, an important feature of SIA is that, besides the existence of
nilpotent infinitesimals (as hypothesized above) it is also consistent
to assume simultaneously the existence of invertible infinitesimals
(of the kind found in nonstandard analysis and useful in analyzing
limits and convergence).  For examples of the development of single-
and multi-variate differential and integral calculus and their
applications to physics in SIA, as well as an outline of the
construction of models, see Bell's book.  The monograph by Moedijk and
Reyes mentioned by Urquhart is

    Ieke Moerdijk & Gonzalo Reyes, Models for Smooth Infinitesimal
    Analysis, Springer Verlag, 1991.

-- 
Todd Wilson                               A smile is not an individual
Computer Science Department               product; it is a co-product.
California State University, Fresno                 -- Thich Nhat Hanh