Timeline for Can we interpret arithmetic in set theory, with exactly PA as the ZFC provable consequences?
Current License: CC BY-SA 4.0
29 events
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| May 26, 2022 at 2:35 | comment | added | Joel David Hamkins | Great ! Thanks. | |
| May 26, 2022 at 2:34 | vote | accept | Joel David Hamkins | ||
| May 26, 2022 at 2:28 | comment | added | Noah Schweber | @JoelDavidHamkins Aha! This was what the shift to $\mathsf{I\Sigma_1}$ was for; I just borked it and then forgot. Fixed! (I am assuming that $\mathsf{PA}$ is $\Sigma_1$-sound, though.) | |
| May 26, 2022 at 2:28 | comment | added | Joel David Hamkins | For example, a model of ZFC can think some nonstandard fragment PA_k of PA is consistent, which would provide an interpretation of PA without the model thinking so. | |
| May 26, 2022 at 2:28 | history | edited | Noah Schweber | CC BY-SA 4.0 |
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| May 26, 2022 at 2:20 | comment | added | Joel David Hamkins | But now I am wondering again about the nonsound case. Why must ZFC prove that the interpretation is a PA interpretation, just because it is? Does this break your argument for the nonsound case? | |
| May 26, 2022 at 2:16 | vote | accept | Joel David Hamkins | ||
| May 26, 2022 at 2:20 | |||||
| May 26, 2022 at 2:12 | comment | added | Noah Schweber | @JoelDavidHamkins Fair enough, I'll delete it. I do think there is an interesting follow-up question re: identifying a better behaved class of interpretations, though. | |
| May 26, 2022 at 2:11 | history | edited | Noah Schweber | CC BY-SA 4.0 |
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| May 26, 2022 at 2:11 | comment | added | Joel David Hamkins | I don't think it is artificial, and indeed I had similar arguments using the universal algorithm, which I realize now achieve the same result as your answer, but I hadn't put it all together. | |
| May 26, 2022 at 2:10 | history | edited | Noah Schweber | CC BY-SA 4.0 |
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| May 26, 2022 at 2:03 | comment | added | Noah Schweber | @JoelDavidHamkins To be fair, I do think the interpretation I've used in the $\Sigma_1$-sound case is silly. But I've changed that to "artificial." (And I've fixed the $\mathsf{I\Sigma_1}$ thing.) | |
| May 26, 2022 at 2:02 | history | edited | Noah Schweber | CC BY-SA 4.0 |
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| May 26, 2022 at 1:57 | comment | added | Joel David Hamkins | For the other side, what seems relevant is that PA$\not\vdash\varphi$ rather than $I\Sigma_1\not\vdash\varphi$. | |
| May 26, 2022 at 1:55 | comment | added | Joel David Hamkins | Ah, now you have mentioned soundness. | |
| May 26, 2022 at 1:54 | comment | added | Joel David Hamkins | I'd prefer a better explanation (for example, you don't appeal to soundness in your argument), and without the perjorative "silly". | |
| May 26, 2022 at 1:53 | history | edited | Noah Schweber | CC BY-SA 4.0 |
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| May 26, 2022 at 1:48 | comment | added | Noah Schweber | @JoelDavidHamkins I've edited the answer. | |
| May 26, 2022 at 1:47 | history | edited | Noah Schweber | CC BY-SA 4.0 |
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| May 26, 2022 at 1:46 | comment | added | Joel David Hamkins | Not sure why the meaning isn't clear. The answer seems to be: Yes, provided ZFC is Sigma_1 sound, otherwise not. If you edit to make this clear, I'll accept. | |
| May 26, 2022 at 1:16 | history | edited | Noah Schweber | CC BY-SA 4.0 |
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| May 26, 2022 at 1:14 | comment | added | Noah Schweber | "It seems you are assuming that that the actual consistency statements are the same as those realized in models of ZFC. Is that right?" I'm not sure what that means. I'm assuming $\mathsf{ZFC}$ is $\Sigma_1$-sound, that's all. | |
| May 26, 2022 at 1:13 | comment | added | Noah Schweber | @JoelDavidHamkins But that trivializes the question in the opposite direction: if $\mathsf{ZFC}$ isn't $\Sigma_1$-sound, then there is some sentence $\varphi$ such that $\mathsf{ZFC}\vdash\mathsf{I\Sigma_1}\vdash\varphi$ but in fact $\mathsf{I\Sigma_1}\not\vdash\varphi$ (shifting attention to $\mathsf{I\Sigma_1}$ to make use of finite axiomatizability here). Then any interpretation of $\mathsf{I\Sigma_1}$ (let alone $\mathsf{PA}$) into $\mathsf{ZFC}$ will have "extra theorems." | |
| May 26, 2022 at 1:13 | comment | added | Joel David Hamkins | It seems you are assuming that that the actual consistency statements are the same as those realized in models of ZFC. Is that right? | |
| May 26, 2022 at 1:08 | comment | added | Joel David Hamkins | Well, that is exactly what is at stake, I would think. | |
| May 26, 2022 at 1:01 | comment | added | Noah Schweber | @JoelDavidHamkins Under reasonable assumptions, every non-$\mathsf{PA}$-theorem is seen to be a non-$\mathsf{PA}$-theorem by some model of $\mathsf{ZFC}$. Or do you not want to assume $\Sigma_1$-soundness? | |
| May 26, 2022 at 0:45 | comment | added | Joel David Hamkins | But perhaps the models of ZFC all think that certain statements are theorems of PA, even though they aren't? It seems you need to know about how internal ZFC theorems relate to the metatheory. | |
| May 26, 2022 at 0:26 | history | edited | Noah Schweber | CC BY-SA 4.0 |
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| May 26, 2022 at 0:14 | history | answered | Noah Schweber | CC BY-SA 4.0 |