FOM: elimination of analytic methods in number theory

Stephen G Simpson simpson at
Wed Feb 25 14:14:59 EST 1998

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
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

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