The Limits of Arithmetical Pluralism: Incompleteness, Large Cardinals, and Graded Non-Pluralism
Year: 2026
Type: preprint
Abstract: Gödelian incompleteness yields arithmetical sentences \(A\) such that \(\mathsf{PA}+A\) and \(\mathsf{PA}+\neg A\) 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 \(\mathsf{PA}+\mathsf{Con}(\mathsf{PA})\) and show that Friedman's concrete incompleteness results pose a serious challenge to non-pluralism about first-order arithmetic. 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.
Source: arXiv (Cornell University)
Author: Cheng, Yong
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214058654