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THE PREHISTORY OF THE SUBSYSTEMS OF SECOND-ORDER ARITHMETIC

Published online by Cambridge University Press:  20 February 2017

WALTER DEAN  and
SEAN WALSH
WALTER DEAN*
Affiliation:
Department of Philosophy, University of Warwick
SEAN WALSH*
Affiliation:
Department of Logic and Philosophy of Science, University of California, Irvine
*
*DEPARTMENT OF PHILOSOPHY UNIVERSITY OF WARWICK COVENTRY CV4 7AL UK E-mail: W.H.Dean@warwick.ac.uk
†DEPARTMENT OF LOGIC AND PHILOSOPHY OF SCIENCE UNIVERSITY OF CALIFORNIA 5100 SOCIAL SCIENCE PLAZA IRVINE, CA 92697-5100 USA E-mail: swalsh108@gmail.com or walsh108@uci.edu
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Abstract

This paper presents a systematic study of the prehistory of the traditional subsystems of second-order arithmetic that feature prominently in the reverse mathematics program promoted by Friedman and Simpson. We look in particular at: (i) the long arc from Poincaré to Feferman as concerns arithmetic definability and provability, (ii) the interplay between finitism and the formalization of analysis in the lecture notes and publications of Hilbert and Bernays, (iii) the uncertainty as to the constructive status of principles equivalent to Weak König’s Lemma, and (iv) the large-scale intellectual backdrop to arithmetical transfinite recursion in descriptive set theory and its effectivization by Borel, Lusin, Addison, and others.

Keywords

01A6003-0303F35history of logicreverse mathematicssecond-order arithmeticpredicativityfinitismconstructive mathematics
Type
Research Article
Information
The Review of Symbolic Logic , Volume 10 , Issue 2 , June 2017 , pp. 357 - 396
Copyright
Copyright © Association for Symbolic Logic 2017 

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References

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