Timeline for Is this theory the complete theory of the real ordered field?
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
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| Mar 24, 2021 at 16:38 | vote | accept | user107952 | ||
| Mar 1, 2021 at 3:04 | answer | added | Alex Kruckman | timeline score: 5 | |
| Mar 1, 2021 at 0:42 | answer | added | Dmytro Taranovsky | timeline score: 17 | |
| Feb 10, 2021 at 20:40 | answer | added | Spencer Dembner | timeline score: 1 | |
| Feb 6, 2021 at 15:13 | history | became hot network question | |||
| Feb 6, 2021 at 15:13 | answer | added | Will Sawin | timeline score: 9 | |
| Feb 6, 2021 at 15:02 | comment | added | Tim Campion | @WillSawin Well, I have one deleted answer where I missed this, you missed it, Ali Enayat's answer missed this, and when the OP originally asked the question in 2015, Asaf Karagila naturally assumed the OP meant "with parameters". Judging by the comments on that question, it appears that the OP themself didn't notice the discrepancy until two years later in 2017! | |
| Feb 6, 2021 at 14:59 | comment | added | Will Sawin | @TimCampion Ah, I see. I missed "without parameters". This question is interesting! | |
| Feb 6, 2021 at 14:17 | answer | added | Emil Jeřábek | timeline score: 4 | |
| Feb 6, 2021 at 3:44 | answer | added | Ali Enayat | timeline score: 4 | |
| Feb 6, 2021 at 3:11 | comment | added | Tim Campion | @MattF. Yeah, the part that's not obvious (and the reason Noah didn't definitively claim it was a counterexample) is whether it does satisfy all the unparameterized LUB axioms. It's confusing to me because if we were working over RCF already, you could say that all the definable sets are just finite unions of intervals (i.e. RCF is o-minimal), and approach what the axioms look like that way. But I don't know how to extend the language of ordered fields to eliminate quantifiers, so you've got to to think about syntactically much more complicated formulas when "enumerating" all the LUB axioms. | |
| Feb 6, 2021 at 2:54 | comment | added | user44143 | So to flesh out Noah's example, consider $\mathbb{R}^{alg}(\pi)$, in the language $(+,-,\times,0,1,\le)$. Unlike $\mathbb{R}$, this has $\neg\forall x \exists y (y^2=x)$, even though it is an ordered field and satisfies all the unparameterized LUB axioms like the LUB for $\{x:x^2<2\}$ or $\{x:x^5+x<1\}$. | |
| Feb 6, 2021 at 2:20 | comment | added | Tim Campion | Also, presumably "real ordered field" is supposed to read "real closed field". | |
| Feb 6, 2021 at 2:15 | comment | added | Tim Campion | In the MSE question, the OP also mentions that they discussed the theory where you allow parameters several years ago in a prequel MSE question, which Asaf Karagila answered. It's worth mentioning that for the current question, Noah Schweber says on MSE he suspects a counterexample is given by taking $\mathbb R^{alg}(\pi)$, say. I'd also suggest comparing to the natural analog of this question in the field language rather than the ordered field language. | |
| Feb 6, 2021 at 2:02 | comment | added | Tim Campion | @WillSawin The issue is that OP's theory only has axioms like that for sets definable without parameters. The sets you're using are defined with parameters -- $a$ in the first case, and the coefficients of $f$ in the second. But in some sense the theory where you allow sets definable with parameters into the scheme is probably the more natural one to use anyway. | |
| Feb 6, 2021 at 2:00 | comment | added | Will Sawin | If we use this characterization of a real closed field: "There is a total order on F making it an ordered field such that, in this ordering, every positive element of F has a square root in F and any polynomial of odd degree with coefficients in F has at least one root in F." then it seems every field satisfying this theory is real closed, since we can take $x^2 < a$ and $f(x)<0$ (if $f(x)$ has odd degree with positive leading term, say) to be definable sets with a least upper bound. | |
| Feb 6, 2021 at 1:50 | history | edited | YCor |
edited tags
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| Feb 6, 2021 at 1:48 | history | edited | Tim Campion | CC BY-SA 4.0 |
Link to MSE crosspost
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| Feb 6, 2021 at 1:35 | comment | added | Tim Campion | In your axiom scheme, do you consider only $\emptyset$-definable sets in one free variable? | |
| Feb 6, 2021 at 0:37 | history | asked | user107952 | CC BY-SA 4.0 |