Timeline for Infinite descending consistency chains
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Jan 14, 2019 at 10:12 | history | edited | Fedor Pakhomov | CC BY-SA 4.0 |
fixed mistakes noticed by Dmiytro Taranovsky
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| Jan 14, 2019 at 10:02 | comment | added | Fedor Pakhomov | @DmytroTaranovsky And you are right about the verification of the fact that a chain of theories is $<_{\mathsf{Con}}$-descending. The theorem in the paper by Walsh and me was that this couldn't be verified in $\mathsf{EA}$ (I don't remember the precise formulations Smorynski and Solovay used). This could be generalize to the result that for any sound r.e. arithmetical $U\supseteq \mathsf{EA}$ and recursive chain $T_i$ of extensions of $U$ it couldn't be the case that $U\vdash \forall x\;\mathsf{Prv}_{T_x}(\mathsf{Con}(T_{x+1}))$. But I don't know how to prove your conjecture. | |
| Jan 14, 2019 at 9:51 | comment | added | Fedor Pakhomov | @DmytroTaranovsky Indeed, you are right the alternative definition of $T_i$'s in the first example, shouldn't be equivalent to the initial one; one would need to use $\mathsf{I}\Sigma_1$ as the base theory to overcome this problem. | |
| Jan 14, 2019 at 9:11 | comment | added | Dmytro Taranovsky | Thank you for the detailed answer. With its help, I added a new answer; let me know if there are gaps in the proof. Also, in the first $T_i$ you give, does "equivalently given as ..." work as EA does not appear to have enough induction to get the least counterexample to the totality of $R_{\mathrm{IΣ}_2}$? Also, near the end of the answer (about the conjecture) "some $T_i$" should be "all $T_i$" (since we can make $T_0$ arbitrarily strong). | |
| Jan 13, 2019 at 17:22 | history | edited | Fedor Pakhomov | CC BY-SA 4.0 |
fixed minor mistake in the proof of 2nd example
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| Jan 13, 2019 at 17:06 | history | answered | Fedor Pakhomov | CC BY-SA 4.0 |