The Limits of Arithmetical Pluralism: Incompleteness, Large Cardinals, and Graded Non-Pluralism

Abstract

Gödelian incompleteness yields arithmetical sentences $A$ such that $PA+A$ and $PA+\neg A$ are both consistent. Are such extensions equally legitimate? I propose graded epistemic arithmetical non-pluralism: justification for choosing between them varies with the set-theoretic strength of $A$. I defend $PA+Con(PA)$ and show Koellner's non-pluralism for first-order arithmetic is inadequate given Friedman's concrete incompleteness. Resolving the selection problem for such sentences turns on justifying large cardinals. Defending graded non-pluralism thus engages Gödel's programme and the justification of very large cardinals.

Accepted and forthcoming in Philosophia Mathematica