Abstract
The title of my paper indicates that I plan to write about foundations for analysis and about proof theory; however, I do not intend to write about the foundations for analysis and thus not about analysis viewed from the vantage point of any “school” in the philosophy of mathematics. Rather, I shall report on some mathematical and proof-theoretic investigations which provide material for (philosophical) reflection. These investigations concern the informal mathematical theory of the continuum, on the one hand, and formal systems in which parts of the informal theory can be developed, on the other. The proof-theoretic results of greatest interest for my purposes are of the following form:
for each F in a class of sentences, F is provable in T if and only if F is provable in T*
where T is a classical set-theoretic system for analysis and T* a constructive theory. In that case, T is called REDUCIBLE TO T*, as the principles of T* are more elementary and more restricted.
Douter de tout ou tout croire, ce sont deux solutions également commodes qui l’une et l’autre nous dispensent de réfléchir.
H. Poincaré
This paper was completed in June 1981; some minor changes were made in August 1982.
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Sieg, W. (1984). Foundations for Analysis and Proof Theory. In: Leblanc, H., Mendelson, E., Orenstein, A. (eds) Foundations: Logic, Language, and Mathematics. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1592-8_10
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