Timeline for Formal systems and interpretations for numbers
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 1, 2020 at 20:29 | comment | added | user4894 | @user21820 Ok I stand educated. Thanks. | |
| Aug 1, 2020 at 17:22 | comment | added | user21820 | @user4894: Oops sorry I did see the year was 2014 but I didn't really think what it meant about our (mathematical) maturity. =) Anyway, basic logic texts do at least assume ZC, because typically we want to prove the completeness and compactness theorems for theories of arbitrary cardinality, and so we need a modicum of set-theoretic strength. Technically, we only need BZC (bounded Zermelo with Choice) for all that (see here) but nobody knows BZC so people just say ZFC. =) | |
| Aug 1, 2020 at 17:14 | comment | added | user4894 | @user21820 Ah, 2014. We were so young and innocent then.Still, I'd be really surprised if basic logic texts are implicitly assuming ZFC. I don't see how that could make sense. | |
| Aug 1, 2020 at 14:37 | comment | added | user21820 | @Casey: Many texts on logic work within a meta-system without ever explicitly stating it. For standard logic texts, you can assume that ZFC suffices if they do not say otherwise. See my answer for more details about the whole spectrum of foundational systems. | |
| Sep 3, 2014 at 16:22 | comment | added | Mauro ALLEGRANZA | I agree with you; my comment is aimed to the question of the OP : "My confusion is that numbers don't exist 'out there' like tangible things do." Neither sets. So every interpretation of number theory must assume that "makes sense" to speak of a universe of astract objects : sets or number. This is the meaning of my distinction between a "mathematically" good answer which is not "philosophically" so good ... | |
| Sep 3, 2014 at 16:21 | comment | added | Casey | @ Mauro ALLEGRANZA. I'm actually not all too concerned with the ontological status of numbers. I had in mind perhaps something like a construction of symbols outside of the formal theory where we can use that as the interpretation. The book seemed to just assume that the numbers exist without any explanation though, so it left me confused. User4894's answer gives me some insight tho; I'll look more into it. (Yours is good as well, lol.) | |
| Sep 3, 2014 at 15:56 | comment | added | user4894 | @MauroALLEGRANZA What would such a proof look like? How can I possibly prove that the number 3 exists? It's an abstract concept, you'd have to say that abstract concepts can exist. You'd never get to the bottom of that. You could write a library of books and never have a full formal proof, because proofs always start with assumptions. | |
| Sep 3, 2014 at 6:30 | comment | added | Mauro ALLEGRANZA | Mathematically good answer ... but, from a "philosophical point of view", if I need a "proof" about the existence of numbers, why their "construction" in a theory which assume the existence of sets can be more ... satisfactory ? | |
| Sep 3, 2014 at 3:14 | history | answered | user4894 | CC BY-SA 3.0 |