Timeline for Do we expect that sufficiently large computable ordinals settle every question of arithmetic?
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| Jan 17 at 0:17 | comment | added | Adam Epstein | @AsafKaragila I knew RM a quarter century ago. | |
| Oct 22, 2022 at 19:34 | comment | added | Timothy Chow | This question is almost a duplicate of another MO question. | |
| Oct 19, 2022 at 18:54 | comment | added | Qiaochu Yuan | @PatrickLutz: yes, that's admittedly a subtlety I didn't quite appreciate when I wrote this question. | |
| Oct 19, 2022 at 16:51 | comment | added | Patrick Lutz | @QiaochuYuan I see. It still seems a bit misleading because most sentences in the language of arithmetic are much more complicated than $\Sigma^0_1$ and some people might believe that $\Sigma^0_1$ sentences have definite truth values while more complicated sentences do not. | |
| Oct 19, 2022 at 16:50 | comment | added | Qiaochu Yuan | @PatrickLutz: "e.g." means "for example," it doesn't mean a bidirectional implication. | |
| Oct 19, 2022 at 16:49 | comment | added | Patrick Lutz | The statement of your question seems to imply that every first order sentence in the language of arithmetic is equivalent (over PA?) to a statement of the form "Turing machine $M$ halts" for some $M$. But this is false (unless PA is inconsistent). The latter is a $\Sigma^0_1$ statement, but not every sentence is equivalent to a $\Sigma^0_1$ sentence. | |
| Oct 17, 2022 at 7:21 | vote | accept | Qiaochu Yuan | ||
| Oct 16, 2022 at 23:25 | comment | added | Asaf Karagila♦ | Ron has some... eccentric... beliefs about the foundations of mathematics. | |
| Oct 16, 2022 at 2:14 | comment | added | Timothy Chow | Maimon mentions Torkel Franzén; I assume he means Franzén's book Inexhaustibility: A Non-Exhaustive Account. I recommend that book highly if you're interested in these sorts of questions. I do agree with Maimon that Hilbert's program is not quite as dead as many make it seem, but I don't think that ordinal analysis is the panacea he seems to make it out to be. Finally, I'd mention that if you look up the literature on "absolute undecidability," you'll find some other interesting ideas about a modern form of Hilbert's program. | |
| Oct 16, 2022 at 2:08 | comment | added | Timothy Chow | I just read Ron Maimon's article, and I find some of the things he says rather misleading. Supposing we take the conjecture you quoted at face value, it still doesn't mean that we can have any realistic hope of "knowing the answer to every [arithmetical] question of mathematics eventually." Even at the level of $\epsilon_0$, we have Noah Schweber on record here as saying that he doesn't regard Gentzen's proof as helping us "know" that PA is consistent if we didn't "know" it already. Also, his comment about iterated consistency statements overstates how far you can get with that idea. | |
| Oct 15, 2022 at 15:11 | comment | added | JoshuaZ | @NoahSchweber Ah! That makes sense! Thank you. | |
| Oct 15, 2022 at 14:30 | comment | added | Noah Schweber | @JoshuaZ Yes: we would need to know ahead of time which computable linear orders are well-ordered in order to rule out false answers. Determining whether a computable linear order is a well-order is much more complicated than the halting problem ($\mathcal{O}$ vs. ${\bf 0'}$). | |
| Oct 15, 2022 at 13:47 | comment | added | JoshuaZ | Seems like this should be false because if it were we could make a Turing machine that determined BB(n) by just dovetailing through all proofs through some axiomatic system along with each axiomatic system adding that alpha_n is well-founded as an additional axiom. Am I missing something? | |
| Oct 15, 2022 at 6:28 | history | became hot network question | |||
| Oct 15, 2022 at 1:31 | answer | added | Dmytro Taranovsky | timeline score: 21 | |
| Oct 15, 2022 at 0:09 | answer | added | Noah Schweber | timeline score: 24 | |
| Oct 14, 2022 at 22:27 | history | asked | Qiaochu Yuan | CC BY-SA 4.0 |