Publikation: Hume's Principle, Bad Company, and the Axiom of Choice
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One prominent criticism of the abstractionist program is the so-called Bad Company objection. The complaint is that abstraction principles cannot in general be a legitimate way to introduce mathematical theories, since some of them are inconsistent. The most notorious example, of course, is Frege’s Basic Law V. A common response to the objection suggests that an abstraction principle can be used to legitimately introduce a mathematical theory precisely when it is stable: when it can be made true on all sufficiently large domains. In this paper, we raise a worry for this response to the Bad Company objection. We argue, perhaps surprisingly, that it requires very strong assumptions about the range of the second-order quantifiers; assumptions that the abstractionist should reject.
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ROBERTS, Sam, Stewart SHAPIRO, 2023. Hume's Principle, Bad Company, and the Axiom of Choice. In: The Review of Symbolic Logic. Cambridge University Press. 2023, 16(4), pp. 1158-1176. ISSN 1755-0203. eISSN 1755-0211. Available under: doi: 10.1017/S1755020322000144BibTex
@article{Roberts2023Humes-57720, year={2023}, doi={10.1017/S1755020322000144}, title={Hume's Principle, Bad Company, and the Axiom of Choice}, number={4}, volume={16}, issn={1755-0203}, journal={The Review of Symbolic Logic}, pages={1158--1176}, author={Roberts, Sam and Shapiro, Stewart} }
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