Timeline for What are some nice uses of ultraproducts/ultrapowers?
Current License: CC BY-SA 4.0
6 events
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| Dec 27, 2023 at 0:41 | comment | added | Asaf Karagila♦ | @KConrad: Yes, I know several other versions of this without ultrapowers. Compactness (for countable languages, to make it choice free), absoluteness arguments to move to $L$ and do it there, which generally isn't technically an ultrapower argument in some cases, or so on. My point in this argument is to show interesting and unusual uses for logic and model theory. Doing it "without" kinda defeats the purpose... | |
| Dec 26, 2023 at 22:11 | comment | added | KConrad | Here is a version of that with no ultraproducts. If $f(x) \in \mathbf Z[x]$ has a rational root $r$, then when $r =a/b \in \mathbf Q$ for integers $a$ and $b$, where $b\not= 0$, we can view $r$ in the localization $\mathbf Z_{(p)}$ whenever $p\nmid b$ and reduce $\mathbf Z_{(p)}$ modulo $p$ to turn $f(r)=0$ in $\mathbf Z_{(p)}$ into $f(r)=0$ in $\mathbf F_p$. Thus an $f(x)$ in $\mathbf Z[x]$ with a root in $\mathbf Q$ has a root in $\mathbf F_p$ for all but finitely many $p$. So when $f(x)$ has no root in $\mathbf F_p$ for infinitely many primes $p$, $f(x)$ also has no root in $\mathbf Q$. | |
| Jul 23, 2023 at 0:07 | comment | added | Noah Schweber | Ooh, I haven't seen this one before - lovely! | |
| Jul 21, 2023 at 18:56 | comment | added | H.C Manu | This is pretty much as outside of the "usual" domain as it gets for an application, very nice answer. | |
| Jul 21, 2023 at 11:17 | history | made wiki | Post Made Community Wiki by Asaf Karagila♦ | ||
| Jul 21, 2023 at 9:45 | history | answered | Asaf Karagila♦ | CC BY-SA 4.0 |