The Prehistory of the Subsystems of Second-Order Arithmetic

TL;DR

This paper explores the historical development of the foundational subsystems of second-order arithmetic, highlighting key influences from mathematicians and logicians that shaped reverse mathematics.

Contribution

It provides a systematic historical analysis of the evolution of subsystems of second-order arithmetic and their foundational significance in reverse mathematics.

Findings

Traces the influence of Poincaré to Feferman on arithmetic definability.

Examines the role of finitism in Hilbert and Bernays' formalizations.

Discusses the uncertainty surrounding the constructive status of Weak K"onig's Lemma.

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 of Friedman and Simpson. We look in particular at: (i) the long arc from Poincar\'e 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 K\"onig'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.