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arXiv · 2506.13524

Extensional Independence

Taishi Kurahashi·Albert Visser
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Abstract

Joel Hamkins asks whether there is a $Π^0_1$-formula $ρ(x)$ such that $ρ(ϕ)$ is independent over ${\sf PA}+ϕ$, if this theory is consistent, where this construction is extensional in $ϕ$ with respect to ${\sf PA}$-provable equivalence. We show that there can be no such extensional Rosser formula of any complexity. We give a positive answer to Hamkins' question for the case where we replace Extensionality by a weaker demand *Consistent Extensionality*. We also prove that we can demand the negation of $ρ$ to be $Π^0_1$-conservative, if we ask for the still weaker *Conditional Extensionality*. We show that an intensional version of the result for Conditional Extensionality cannot work.

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Taishi Kurahashi, Albert Visser. 2026-02-26. Extensional Independence. https://arxiv.org/abs/2506.13524

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