Σ01 soundness isn’t enough: Number theoretic indeterminacy’s unsavory physical commitments

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Abstract
It’s sometimes suggested that we can (in a sense) settle the truth-value of some statements in the language of number theory by stipulation, adopting either φ or ¬φ as an additional axiom. For example, in Clarke-Doane (2020b) and a series of recent APA presentations, Clarke-Doane suggests that any Σ01 sound expansion of our current arithmetical practice would express a truth. In this paper, I’ll argue that (given a certain popular assumption about the model-theoretic representability of languages like ours) we can’t know ourselves to have any such freedom.
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Archival date: 2021-07-10
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2021-04-16

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