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arXiv · 2609.22797

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

Yong Cheng
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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.

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BibTeXRIS

Yong Cheng. 2026-10-22. The Limits of Arithmetical Pluralism: Incompleteness, Large Cardinals, and Graded Non-Pluralism. https://arxiv.org/abs/2609.22797

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