Timeline for Construction of model of arithmetic from an arbitrary model
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Jun 22, 2016 at 10:30 | comment | added | Erfan Khaniki | @JoelDavidHamkins:What is the name of Gaifman's paper that you used in your comment? | |
| Jun 21, 2016 at 17:03 | comment | added | Emil Jeřábek | I am no expert on minimal extensions, but I am skeptical the theorem could have a meaningful generalization to nonelementary extensions, of the standard model in particular. For instance, every countable (?) nonstandard model of PA (or even $IE_1$) has arbitrarily short initial nonstandard submodels that are models of PA (or of any r.e. $\Sigma_1$-sound theory extending $I\Sigma_1$). | |
| Jun 21, 2016 at 16:50 | comment | added | Joel David Hamkins | @EmilJeřábek Do you know if the minimal extension theorem of Gaifman and related results is only concerned with minimal elementary extensions, or could it also be used to provide negative instances of the phenomenon here? | |
| Jun 21, 2016 at 13:23 | history | edited | Erfan Khaniki | CC BY-SA 3.0 |
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| Jun 21, 2016 at 13:16 | history | edited | Erfan Khaniki | CC BY-SA 3.0 |
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| Jun 21, 2016 at 12:58 | comment | added | Emil Jeřábek | The paper of Kaye I meant was Diophantine induction, Annals of Pure and Applied Logic 46 (1990), no. 1, pp. 1–40, and the followup paper Hilbert’s tenth problem for weak theories of arithmetic, Annals of Pure and Applied Logic 61 (1993), no. 1–2, 63–73 (see also mathoverflow.net/a/168412). | |
| Jun 21, 2016 at 12:32 | comment | added | Erfan Khaniki | @EmilJeřábek: Where can I find the Kaye's paper that you mentioned? | |
| Jun 21, 2016 at 12:29 | comment | added | Erfan Khaniki | @EmilJeřábek: The order of quantifiers in my question is: For every nonstandard model $M$ of PA, for every nonstandard element $a$, For which theories $T$, there exists $M'$... Can you gather your comments and post an answer with more details? (what do you mean by definable by a $\Delta_0$-formula in $M$, or formalized MRDP) | |
| Jun 21, 2016 at 10:47 | comment | added | Joel David Hamkins | @Emil, I had just wanted you to clarify your comment, since you referred to a definable nonstandard element of $M$, but some models have no such elements, and so this depends on $M$. I agree with you that the quantifiers in the question are not clear. | |
| Jun 21, 2016 at 8:51 | comment | added | Emil Jeřábek | Meanwhile, still using my original reading: (1) it is easy to that the property holds for PA^- and for Z-rings. I think the argument will generalize to IOpen with some care, but I haven't worked out the details. (2) If one could show the property for IE_1, this would imply it doesn't prove the MRDP theorem, which is an open problem. While the answer is almost certainly that IE_1 doesn't prove MRDP, I find it conceivable that one might refute the property for IE_1 directly using Kaye's methods for showing IE_1 Diophantine undecidable, but this is only a guess. | |
| Jun 21, 2016 at 8:28 | comment | added | Emil Jeřábek | @Joel I read the question as "for which theories T is it true that: for every model M of PA, for every nonstandard a in M, there exists M' etc." So, my argument works as long as there exists at least one model of PA with a nonstandard $\Delta_0$-definable element, which it does. The existence of other models without such elements is irrelevant. | |
| Jun 21, 2016 at 1:12 | comment | added | Joel David Hamkins | But meanwhile, the theory TA of true arithmetic does not have the desired property, because of Gaifman's theorem on minimal extensions. That is, there is a minimal extension of the standard model, which would violate the desired properties. | |
| Jun 21, 2016 at 0:32 | comment | added | Joel David Hamkins | @EmilJeřábek Could you explain your comment? In a nonstandard model of true arithmetic, for example, which extends that theory, there are no such definable nonstandard elements. | |
| Jun 21, 2016 at 0:05 | comment | added | Erfan Khaniki | @EmilJeřábek: Thank you for your comment. What about weaker theories like $I\Delta_0$ or even weaker like $IE_1$? (I edited my post as you suggested.) | |
| Jun 21, 2016 at 0:02 | history | edited | Erfan Khaniki | CC BY-SA 3.0 |
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| Jun 20, 2016 at 22:50 | comment | added | Emil Jeřábek | If you take for $a$ a (parameter-free) $\Delta_0$-definable element of $M$, the conditions imply that $M'$ is not a $\Delta_0$-elementary submodel of $M$, hence it cannot satisfy the formalized MRDP theorem. That is, this property is false for any theory $T\supseteq I\Delta_0+\mathit{EXP}$. | |
| Jun 20, 2016 at 22:07 | history | asked | Erfan Khaniki | CC BY-SA 3.0 |