Timeline for Defining the standard model of PA so that a space alien could understand
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 24, 2019 at 16:40 | comment | added | Pace Nielsen | Noah, I'm trying to understand your "dynamic approximation" idea. What would that look like in practice? What do you mean by "hold a model"? | |
| May 23, 2019 at 20:17 | comment | added | Timothy Chow | @PaceNielsen : If you're complaining about the choice of the word "standard," I don't think it's any different from the way the adjective "normal" is overused in mathematics. What's "normal" about the "normal distribution"? Or is your complaint that if something can't be defined non-circularly then we can't know what it is? But in that case, I'd claim that the number "3" can't be defined non-circularly either, yet I don't believe we can't know what "3" is. | |
| May 23, 2019 at 17:58 | comment | added | abo | @PaceNielsen. I guess they talk about it because in a context such as ZF, or some such theory, it has a meaning (the one Monroe is talking about). That is, for mathematicians working in the context of ZF, it makes sense to talk about the standard model, and it helps them to do mathematics.This doesn't provide a "definition" in your sense, however. | |
| May 23, 2019 at 16:57 | comment | added | Monroe Eskew | I think we can be absolutist about the notion “smallest inductive set.” It’s true that this notion can be interpreted differently in different models of set theory, but the concept of algebraic generation is so primitive and clear. Further, the standard numbers always embed into any nonstandard version, so there is some absolute, inter-universal sense in which it is “smallest.” | |
| May 23, 2019 at 16:40 | comment | added | Pace Nielsen | @abo If that's the case, then why do people still talk about a "standard" model? As a rhetorical device? If so, it is highly misleading. (And I'm not being flippant either.) | |
| May 23, 2019 at 15:24 | comment | added | abo | It seems to me to be pretty clear you can't define the natural numbers, in any reasonable sense, or in any sense agreeable to you, in a non-circular way. So - next question? I hope I'm not seeming to be flippant here, but still. | |
| May 23, 2019 at 14:14 | comment | added | Pace Nielsen | By "expressible" I meant definable in a non-circular way. Your comment above, about understanding "only finitely many" only after having access to the natural numbers gets to the heart of it. | |
| May 23, 2019 at 5:16 | history | answered | Noah Schweber | CC BY-SA 4.0 |