FOM: elimination of analytic methods in number theory

[Stephen G Simpson]

Joe Shipman 24 Feb 1998 15:08:27 writes:
 > the apparent necessity of analytic methods in number theory [are
 > these always eliminable?]).

An answer to this question is in my paper "Partial Realizations of
Hilbert's Program" at http://www.math.psu.edu/simpson/papers/hilbert/.
As remarked there, the results there imply that analytic methods can
be eliminated wholesale in favor of "elementary" number-theoretic
methods, if by "elementary" we mean formalizable in PRA (= primitive
recursive arithmetic).  This is better than the Feferman/Takeuti
result cited by Martin Davis 24 Feb 1998 15:37:22, because PRA is much
weaker than PA (= first-order Peano arithmetic).  PRA is an embodiment
of Hilbert's finitism, while PA goes beyond finitism; see W. W. Tait,
Finitism, Journal of Philosophy, 1981, pp. 524-546.

-- Steve