FOM: Peano and AC

[Walter Felscher]

The statement that infiniteley many choices cannot be made (i.e. that 
an AC would be false) was made by Peano 1890 on p.210 of his

   D/emonstration de l'int/egrabilit/e des /equations diff/erentielles
   ordinaires.  Math.Ann. 37 (1890) 182-229 .

Already in his

   Sull' integrabilit\a della equazioni differenziali di primo ordine.
   Atti Acad.Sci.Torino 21 (18856/86) 677-685

Peano has shown what now is called _Peano's exsistence theorem_ :

   Let f be function continuous on a closed rectangle R of the real
   plane. For every point (x,y) of R there exists a real function h ,
   defined and differentiable in a neighbourhood U of x such that

     h(x)=y  ,  h'(u)=f(u,h(u))  for all u in U .

In his later article Peano extended this to systems of differential
equations. While in most of today's presentations Peano's theorem
is proved with help of the Arzela-Ascoli compactness principle for
function sequences, Peano's construction of a solution h proceeds as
follows. Consider a suitable interval U=(a,b) around x . For the 
abscissae c(n,i) = a + i.(b-a)/2^n , 0<i<2^n , 0<n<omega , define by
recursion on n approximate solutions which give rise to non empty sets
A(n,i) (constructed with help of Cantor's intersection theorem). Choosing
the value of h at c(n,i) in the set A(n,i) , Peano shows that h is
uniquely determined also in the remaining points of U and is solution of
the differential equation on U . About these choices Peano writes l.c.

   Mais comme on ne peut pas appliquer une infinit/e de fois une loi
   arbitraire avec laquelle \a une classe <alpha> on fait correspondre un
   individu de cette classe, on a form/e ici une loi d/etermion/ee avec 
   laquelle \a chaque classe <alpha>, sous des hypth\eses convenables,
   on fait correspondre un individu de cette classe. 

The well-defined rule of choice, which Peano mentions here, can be defined
because the sets A(n,i) turn out to be closed such that one may take the
supremum of each of them; in the case of systems of differential equations
the sets A(n,i) are in n-space and the suprema ar taken coordinate-wise.  

So it was 14 years before Zermelo formulated AC that Peano thought of such
principle and rejected it. And not only that: he had the sharpness of mind
to replace, in his situation, an arbitrary choice by one that could be
defined explicitly, in view of the particular nature of the sets he chose
from.

All of this I observed long ago on p.112 of my German book "Naive Mengen
und Abstrakte Zahlen", vol.3 , Mannheim 1979 .


W.F.