Timeline for Defining the standard model of PA so that a space alien could understand
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 23, 2019 at 17:19 | comment | added | Rodrigo Freire | Yes, that is about it. The metamathematical numbers are not a model, therefore neither standard nor nonstandard. Models are mathematical objects living inside a mathematical theory (of sets) and modeling formal systems like PA. The metamathematics is not like that. | |
| May 23, 2019 at 17:13 | comment | added | Pace Nielsen | @RodrigoFreire I could go with that answer. There would be a "standard model of PA" (in your metamathematical setup) but no "standard model of the natural numbers" because the natural numbers would be prior to the modeling process. | |
| May 23, 2019 at 17:03 | comment | added | Rodrigo Freire | @PaceNielsen So what you call actual numbers can be identified with the metamathematical numbers (the numbers you need to talk about formal systems) and the priority question does not apply. | |
| May 23, 2019 at 16:43 | comment | added | Pace Nielsen | @MonroeEskew See the second half of Noah's answer for problems with this approach. | |
| May 23, 2019 at 16:27 | comment | added | Monroe Eskew | @PaceNielsen Smallest collection containing zero and closed under successor. | |
| May 23, 2019 at 14:52 | comment | added | Pace Nielsen | Precisely. What are the actual numbers? You apparently need them to talk about formal systems, and metamathematical numbers. I might say "things that count". The aliens might say "the possible steps taken by a supertask machine" (or in other words, the numbers $n$ such that a supertask machine says that $\exists n...$ gives a positive answer). The aliens might ask us how our notion differs from theirs. Could we express how they differ? (Do they differ?) | |
| May 23, 2019 at 14:45 | comment | added | Rodrigo Freire | What are the actual numbers? | |
| May 23, 2019 at 14:42 | comment | added | Rodrigo Freire | Yes, as long as metamathematics is prior to formal systems. There is no formal system such as PA without metamathematics, at least in the usual hilbertian conception. | |
| May 23, 2019 at 14:36 | comment | added | Pace Nielsen | Are metamathematical numbers somehow prior to actual numbers? Seems circular to me. | |
| May 23, 2019 at 14:32 | comment | added | Rodrigo Freire | I am using SS...S0 as a numeral, a term of PA determined by a metamathematical number (the metamathematical number of S’s). | |
| May 23, 2019 at 14:28 | comment | added | Pace Nielsen | But "SSS...S0" is not actually a numeral. To define "numeral" don't you already need access to the natural numbers, thus making the definition of the standard model circular? | |
| May 23, 2019 at 14:24 | history | answered | Rodrigo Freire | CC BY-SA 4.0 |