On the philosophical meaning of reverse mathematics. (English) Zbl 0878.00006

Czermak, Johannes (ed.), Philosophy of mathematics. Proceedings of the 15th international Wittgenstein-Symposium, August 16-23, 1992, Kirchberg am Wechsel, Austria. Part I. Wien: Hölder-Pichler-Tempsky. Schriftenreihe der Wittgenstein-Gesellschaft. 20/I, 173-184 (1993).
Hilbert’s program was created to save all of the integrity of classical mathematics dealing with actual infinity. It was struck by Gödel’s incompleteness results, although it cannot be answered definitely, as the author points out (p. 175), whether it was rejected by these results since “Hilbert’s program was not formulated precisely enough.”
The author discusses two strategies to overcome the effect of Gödel’s results: (1) the idea to extend “the admissible methods and allowing general constructive methods instead of finitistic ones only” (p. 176) which leads to a “generalized Hilbert’s program”; (2) the attempt to deal with the question “how much of infinitistic mathematics can be developed within formal systems which are conservative over finitistic mathematics with respect to real sentences” (p. 177) which leads to “relativized versions” of the program. A prominent example of the second strategy is the “reverse mathematics” created by Friedman and Simpson. The author especially discusses to what extent the second order arithmetic \(A^-_0\) and its fragments \(RCA_0\), \(WKL_0\), and \(ACA_0\) provide partial realizations of Hilbert’s program. He furthermore hints at logical and mathematical applications of these systems.
For the entire collection see [Zbl 0836.00022].


00A30 Philosophy of mathematics
01A60 History of mathematics in the 20th century
03F03 Proof theory in general (including proof-theoretic semantics)
03F35 Second- and higher-order arithmetic and fragments