Timeline for What is the proof-theoretic ordinal of bare $\mathsf{NFU}$?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Apr 2, 2021 at 1:09 | comment | added | Ali Enayat | @JamesHanson $\mathsf{NFU}^{-\infty}$ is known to prove that the strongly cantorian cardinals satisfy $\mathrm{I}\Delta_0 + \mathsf{Exp}$ plus $\mathrm{B}\Sigma_1$ (the collection scheme for $\Sigma_1$ formula). Surprising, they also satisfy even more, namely, for a specific definable cut $\mathsf{I}(x)$ of strongly Cantorian numbers, $\mathsf{Con} ({\mathrm{I}\Delta_0 + \mathsf{Exp}})$ holds in $\mathsf{I}$, as shown by Solovay (alas, again unpublished). The precise theory that holds in the strongly Cantorian numbers (provably in $\mathsf{NFU}^{-\infty}$) has not been identified. | |
| Apr 2, 2021 at 0:49 | comment | added | Ali Enayat | @NoahSchweber About Solovay's proofs: they were communicated directly to me via several long emails directly to me in 2002. | |
| Apr 2, 2021 at 0:47 | comment | added | Ali Enayat | @NoahSchweberThe only published proof of the consistency of $\mathsf{NFU}^{- \infty}$ is the one that is implicit in Jensen's original paper on $\mathsf{NFU}$. Indeed, to my knowledge, the only published paper that explicitly talks about $\mathsf{NFU}^{- \infty}$ is the paper (available: fs2.american.edu/enayat/www/Enayat-Tehran-August%2024.pdf) which was published in Logic in Tehran, Lecture Notes in Logic, vol. 26, Association for Symbolic Logic, 2006. It shows the intimate relationship between Peano Arithmetic and a natural strengthening of $\mathsf{NFU}^{- \infty}$). | |
| Apr 1, 2021 at 23:13 | vote | accept | James E Hanson | ||
| Apr 1, 2021 at 23:13 | comment | added | James E Hanson | Thank you. Is $\mathsf{NFU}^{-\infty}$ known to be conservative over $\mathrm{I}\Delta_{0} + \mathsf{Exp}$ (if we think of the finite strongly Cantorian cardinals as the naturals)? | |
| Apr 1, 2021 at 23:06 | comment | added | Noah Schweber | Are there published proofs of (1)-(2) with a stronger system in place of the weak "base theory" $\mathsf{I\Delta_0+(Sup)Exp}$? And are Solovay's arguments available somewhere, even if unpublished? | |
| Apr 1, 2021 at 23:02 | history | edited | Ali Enayat | CC BY-SA 4.0 |
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| Apr 1, 2021 at 20:53 | history | answered | Ali Enayat | CC BY-SA 4.0 |