What reverse mathematics teaches about foundations
PAPER VI — PHILOSOPHY · QUESTION 9 OF 30
In plain terms: The free entry on mathematics run backwards — asking of each theorem exactly which axioms it secretly demands.
The question as set
What, if anything, does reverse mathematics teach us about foundations?
The question in Paper VI · Its section on the reference page
The question in context
This is question 9 of the 30 set in Paper VI — Philosophy of The Truly Hardest Exam in the World, the examination of The College of All Minds. In the College’s subject index it is filed under Foundations of mathematics. Like every question of the College, it carries three references of record, verified live at publication and described twice — technically, and in plain terms — so the ground can be judged before it is walked. The response required is an argument, not a survey, of at most three thousand words, as technical as the question demands and no more.
The references of record
The programme entire: calibrating theorems against the Big Five subsystems, and what the calibration means philosophically.
In plain terms: The free entry on mathematics run backwards — asking of each theorem exactly which axioms it secretly demands.
The technical apparatus indexed: subsystems of second-order arithmetic and the equivalences that define the hierarchy.
In plain terms: A free working reference for the machinery — the ladder of systems on which theorems are placed.
The foundational stakes: reverse mathematics as the instrument measuring how much of Hilbert's aim survives Gödel.
In plain terms: The free account of the grand project whose partial salvage is precisely what the calibrations reveal.
Background: Reverse mathematics · Second-order arithmetic