Timeline for Ultraconsistency & Truth
Current License: CC BY-SA 3.0
29 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Dec 23, 2017 at 16:08 | comment | added | Emil Jeřábek | Any $\Sigma_1$-sound theory is ultraconsistent. (This is likely if and only if.) Thus, 1. there are ultraconsistent theories T, H whose union is (plain) inconsistent, and 3. there is an ultraconsistent theory that's not even $\Sigma_2$-sound. 2. is likely easily shown true if you fix issues with the formalization mentioned in Gro-Tsen's comment. | |
| Dec 23, 2017 at 15:18 | history | reopened |
Gro-Tsen paul garrett Jan-Christoph Schlage-Puchta Stefan Kohl♦ Johannes Hahn |
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| Dec 20, 2017 at 17:07 | comment | added | Zuhair Al-Johar | @Gro-Tsen Thanks for drawing my attention to making the explicit note. I made that note now. | |
| Dec 20, 2017 at 17:05 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| Dec 20, 2017 at 16:20 | comment | added | user21820 | @Gro-Tsen: Your last comment makes me wonder whether you perhaps know the answer to my question here? =) | |
| Dec 20, 2017 at 14:41 | comment | added | Gro-Tsen | Also, I don't know the answers to your (modified) questions, but I suspect "ultraconsistency" is strongly tied to $\Sigma_2$-soundness. A candidate for a theory that would be "ultraconsistent" but not sound (I suppose "FALSE" means unsound) might be the theory $\mathrm{PA}$+"$\mathrm{PA}$ is $\Sigma_2$-unsound", which I think is $\Sigma_2$-sound (but obviously not sound), and I suspect "ultraconsistent". | |
| Dec 20, 2017 at 14:38 | comment | added | Gro-Tsen | Now that you've edited the question to iterate along a recursive notation system, it makes sense, so I nominated for reopening. But it would have been better if you had made it clear in your edit that your original question was wrong and that you fixed it, so people can see it and vote for reopening. | |
| Dec 20, 2017 at 14:05 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| Dec 20, 2017 at 7:54 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| Dec 20, 2017 at 7:32 | review | Reopen votes | |||
| Dec 23, 2017 at 15:18 | |||||
| Dec 20, 2017 at 7:24 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
I've responded to the Hold objection, by improving my question.
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| Dec 20, 2017 at 7:13 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| Dec 20, 2017 at 5:23 | history | undeleted | Zuhair Al-Johar | ||
| Dec 20, 2017 at 2:06 | history | deleted | Zuhair Al-Johar | via Vote | |
| Dec 19, 2017 at 23:57 | history | closed |
Andrés E. Caicedo Johannes Hahn Stefan Kohl♦ Nik Weaver Pace Nielsen |
Needs details or clarity | |
| Dec 18, 2017 at 20:34 | comment | added | Andrés E. Caicedo | (@Zuhair More likely, I think you did not understand the problem Gro-Tsen mentioned.) | |
| Dec 18, 2017 at 20:33 | comment | added | Andrés E. Caicedo | @Zuhair Are you saying that you have a way of circumventing the issues pointed out in mathoverflow.net/a/67237/6085 ? | |
| Dec 18, 2017 at 20:30 | comment | added | Gro-Tsen | If you work in second-order logic (with full models), then I don't know what $\mathrm{Con}(T)$ means, but $\mathrm{PA}^2$ is categorical (meaning, has a single model, the intended model) from the start, so adding axioms to it is unlikely to produce anything interesting. | |
| Dec 18, 2017 at 20:22 | comment | added | Zuhair Al-Johar | @Gro-Tsen, it is well defined in the second order language of arithmetic. | |
| Dec 18, 2017 at 20:18 | comment | added | Gro-Tsen | This is something of a FAQ (and pretty much everyone makes this mistake at some point): $\mathrm{Con}(T)$ does not depend merely on the set of axioms of $T$ but on the way they are enumerated. So $T_i$ is not well-defined, unless you let $i$ be a recursive ordinal notation and not merely a recursive ordinal (but then you run into other problems). | |
| Dec 18, 2017 at 20:17 | comment | added | Zuhair Al-Johar | Thanks. I've responded to the basic requirements of T, it should be an extension of PA. | |
| Dec 18, 2017 at 20:10 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| Dec 18, 2017 at 19:47 | review | Close votes | |||
| Dec 19, 2017 at 23:57 | |||||
| Dec 18, 2017 at 19:38 | comment | added | Andrés E. Caicedo | I tried to change the format into something palatable. You may want to explain what basic requirements you are imposing on your theories $T$. Is $T_0$ a subtheory (or an extension) of $\mathsf{PA}$? Is it a theory in an arbitrary language but capable of coding enough recursion to make sense of the $T_i$ and of consistency statements? If the latter, then you need to specify what the natural model is that you have in mind, to make sense of the notion of "false". | |
| Dec 18, 2017 at 19:35 | history | edited | Andrés E. Caicedo | CC BY-SA 3.0 |
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| Dec 18, 2017 at 19:28 | comment | added | Andrés E. Caicedo | What a horror... | |
| Dec 18, 2017 at 19:24 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| Dec 18, 2017 at 19:14 | history | asked | Zuhair Al-Johar | CC BY-SA 3.0 |