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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+¬ 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 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.
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BibTeX
@misc{cheng2026limits,
title = {{The Limits of Arithmetical Pluralism: Incompleteness, Large Cardinals, and Graded Non-Pluralism}},
author = {Yong Cheng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.22797},
eprint = {2609.22797},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22797},
}
RIS
TY - UNPB
TI - The Limits of Arithmetical Pluralism: Incompleteness, Large Cardinals, and Graded Non-Pluralism
AU - Yong Cheng
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2609.22797
UR - https://arxiv.org/abs/2609.22797
AB - Gödelian incompleteness yields arithmetical sentences A such that PA+A and PA+¬ 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 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.
ER -