Foundations for Analysis and Proof Theory

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.

[1]  Solomon Feferman,et al.  Inductive definitions and subsystems of analysis , 1981 .

[2]  E. Zermelo.  Grundlagen einer allgemeinen Theorie der mathematischen Satzsysteme , 1935 .

[3]  A. Fraenkel.  Untersuchungen über die Grundlagen der Mengenlehre , 1925 .

[4]  L. Brouwer.  Über Definitionsbereiche von- Funktionen , 1927 .

[5]  Paul Bernays.  BEMERKUNGEN ZU LORENZEN'S STELLUNGNAHME IN DER PHILOSOPHIE DER MATHEMATIK , 1978 .

[6]  Gregory H. Moore.  Beyond first-order logic: the historical interplay between mathematical logic and axiomatic set theory , 1980 .

[7]  G. Takeuti.  Two Applications of Logic to Mathematics , 1978 .

[8]  J. Heijenoort.  From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 , 1967 .

[9]  R. O. Gandy.  Bertrand Russell, as Mathematician , 1973 .

[10]  Paul Bernays.  Hilberts Untersuchungen über die Grundlagen der Arithmetik , 1935 .

[11]  Georg Kreisel,et al.  A survey of proof theory , 1968, Journal of Symbolic Logic.

[12]  Solomon Feferman.  A MORE PERSPICUOUS FORMAL SYSTEM FOR PREDICATIVITY , 1978 .

[13]  D. Hilbert,et al.  Die Grundlegung der elementaren Zahlenlehre , 1931 .

[14]  Solomon Feferman,et al.  Systems of predicative analysis , 1964, Journal of Symbolic Logic.

[15]  Paul Bernays,et al.  Abhandlungen zur Philosophie der Mathematik , 1977 .

[16]  E. Zermelo.  Untersuchungen über die Grundlagen der Mengenlehre. I , 1908 .

[17]  Harvey M. Friedman.  Iterated Inductive Definitions and Σ21-AC , 1970 .

[18]  K. Schutte.  Review: Paul Bernays, Die Philosophie der Mathematik und die Hilbertsche Beweistheorie , 1978 .

[19]  Akiko Kino,et al.  Intuitionism and Proof Theory , 1970 .

[20]  W. Buchholz.  Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-theoretical Studies , 1981 .

[21]  É. Borel.  Leçons sur la théorie des fonctions , 2009 .

[22]  D. Hilbert,et al.  Über homogene Functionen. , 1900 .

[23]  David Hilbert,et al.  Grundlagen der Geometrie , 2022 .

[24]  P. Bernays,et al.  Grundlagen der Mathematik , 1934 .

[25]  S. Feferman.  Formal Theories for Transfinite Iterations of Generalized Inductive Definitions and Some Subsystems of Analysis , 1970 .

[26]  B. Russell.  Mathematical Logic as Based on the Theory of Types , 1908 .

[27]  Paul Bernays.  Über Hilberts Gedanken zur Grundlegung der Arithmetik. , 1922 .

[28]  Solomon Feferman,et al.  Theories of Finite Type Related to Mathematical Practice , 1977 .

[29]  William W. Tait.  Functionals Defined by Transfinite Recursion , 1965, J. Symb. Log..

[30]  D. Hilbert,et al.  Probleme der Grundlegung der Mathematik , 1930 .

[31]  G. Gentzen.  Untersuchungen über das logische Schließen. I , 1935 .

[32]  K. Gödel.  Philosophy of mathematics: Russell's mathematical logic , 1984 .