FOM: elimination of analytic methods in number theory

[Stephen G Simpson]

Olivier Gerard writes:

 > "Le theoreme de Dirichlet est finitiste"  Patrick CEGIELSKI,
 > Jussieu IBP-LITP Internal Report  92.90  (May 1992)
 > 
 > and is certainly available electronically (I can lookup if anyone is

Ah, good!  Could you please post the exact references to Cegielski's
paper, including the electronic one.  Does Cegielski do anything with
the prime number theorem?  Other theorems of number theory?

 > I remember he uses Parsons work on Sigma_1 induction.

This fits with what I was saying about Dirichlet's theorem.  Parsons'
result is that Sigma_1 induction is conservative over PRA for Pi^0_2
sentences.  Sigma_1 induction is known to be just the first order part
of WKL_0.

On the other hand, the nice thing about WKL_0 is that it supports a
great deal of elementary real and complex analysis, so it's
illuminating with respect to the question of eliminating complex
function theory in favor of elementary methods.  From your description
of Cegielski's work, it appears that Cegielski is not taking this
analytic approach, rather he is working directly with Sigma_1
induction.  But let me look at his paper before I comment further.

-- Steve