Extensional Independence
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908868609572864 |
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| author | Kurahashi, Taishi Visser, Albert |
| author_facet | Kurahashi, Taishi Visser, Albert |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_13524 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extensional Independence Kurahashi, Taishi Visser, Albert Logic 03F30, 03F40 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. |
| title | Extensional Independence |
| topic | Logic 03F30, 03F40 |
| url | https://arxiv.org/abs/2506.13524 |