Timeline for A "surnatural numbers" as a largest model of the natural numbers
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Apr 4, 2022 at 21:17 | comment | added | nombre | @MikeBattaglia Yes, if you say non-standard model of $\mathbb{N}$, then people will understand that you are talking about true arithmetic. | |
| Apr 4, 2022 at 21:09 | comment | added | Mike Battaglia | I think I've been confused about this terminology all this time! When we take an ultrapower of the naturals, for instance, to form the hypernaturals, and we say this is a nonstandard model of the naturals, are we then saying that it's a nonstandard model of "true arithmetic" i.e. stronger than just a nonstandard model of PA? | |
| Apr 4, 2022 at 20:44 | comment | added | nombre | @MikeBattaglia Yes, PA usually refers to the recursively enumerated theory consisting in Roinson arithmetic + the induction scheme, whereas "true arithmetic" is sometimes used to refer to the first order theory of $\mathbb{N}$. | |
| Apr 4, 2022 at 19:40 | comment | added | Mike Battaglia | Thanks @nombre, the references to integer parts of real-closed fields looks very useful and I will read that. I am not quite following the first part though, where you are saying "it is unlikely that one could give a nice description of such an object within the class 𝐍𝐨 of surreal numbers" and also saying "An easier task would be to try to identify models of Peano Arithmetic (PA) within 𝐍𝐨." Aren't these the same thing? Or are you somehow drawing a distinction between the first-order theory of $\Bbb N$ as a semi-ring vs the first-order theory of PA? | |
| Mar 21, 2022 at 20:15 | history | answered | nombre | CC BY-SA 4.0 |