Timeline for Do we expect that sufficiently large computable ordinals settle every question of arithmetic?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Oct 17, 2022 at 7:21 | vote | accept | Qiaochu Yuan | ||
| Oct 15, 2022 at 21:12 | comment | added | Timothy Chow | @QiaochuYuan I think that "not being tautological" is a reference to the concept of ordinal analysis. So for example, one could argue that "well-foundedness of the Feferman–Schütte ordinal settles every theorem of $\mathsf{ATR}_0$." This is basically what Dmytro Taranovsky is addressing in his answer. | |
| Oct 15, 2022 at 17:49 | comment | added | Qiaochu Yuan | Thanks very much for the clarification, this is helpful. So one would somehow like a computable order $\alpha$ which is "not tautologically related to $\varphi$," which one can stare at on its own terms to build intuition about whether one believes that $\varphi$ is true or false? I seem to recall a similar situation came up in one of Scott Aaronson's questions about the proof-theoretic ordinal of ZF. Can we make precise this notion of $\alpha$ "not being tautological"? (The tautological version is already interesting to me, for now!) | |
| Oct 15, 2022 at 3:12 | comment | added | Noah Schweber | @QiaochuYuan See my edit. | |
| Oct 15, 2022 at 3:12 | history | edited | Noah Schweber | CC BY-SA 4.0 |
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| Oct 15, 2022 at 0:46 | comment | added | Qiaochu Yuan | Thanks! Can you elaborate a little more on that last sentence? What's artificial about this to you and what would a less artificial version of this idea (speaking very vaguely) look ilke? | |
| Oct 15, 2022 at 0:09 | history | answered | Noah Schweber | CC BY-SA 4.0 |