Timeline for Undefinability of $\mathbb{Z}$ in the reals
Current License: CC BY-SA 3.0
18 events
| when toggle format | what | by | license | comment | |
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| Apr 24, 2017 at 18:28 | comment | added | Emil Jeřábek | @NateEldredge The statement that if it were definable, it could be written as a finite union of solution sets of systems of polynomial inequalities, is correct, but nontrivial. Indeed, this is exactly the Tarski’s theorem on quantifier elimination mentioned in the question. | |
| Apr 24, 2017 at 18:20 | comment | added | Nate Eldredge | This is really naive and probably wrong - but if it were definable, wouldn't you be able to write $\mathbb{Z}$ as a finite union of solution sets of systems of polynomial inequalities? So this is sounding like an algebraic geometry statement. But everything I know about this I just learned from en.wikipedia.org/wiki/Semialgebraic_set | |
| Apr 24, 2017 at 18:13 | history | edited | Mohammad Golshani | CC BY-SA 3.0 |
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| Apr 24, 2017 at 17:21 | comment | added | Will Sawin | @MohammadGolshani How can you hope to work with the structure of some formula without eliminating quantifiers from that formula and thus proving quantifier elimination? | |
| S Apr 24, 2017 at 17:14 | history | suggested | Erfan Khaniki | CC BY-SA 3.0 |
the link is unrelated to the topic.
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| Apr 24, 2017 at 16:49 | review | Suggested edits | |||
| S Apr 24, 2017 at 17:14 | |||||
| Apr 24, 2017 at 14:49 | answer | added | Cubikova | timeline score: 7 | |
| Apr 24, 2017 at 14:42 | comment | added | Mohammad Golshani | Yes, I meant multiplication, I edited it, but it is interesting to know the simpler proof. | |
| Apr 24, 2017 at 14:41 | history | edited | Mohammad Golshani | CC BY-SA 3.0 |
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| Apr 24, 2017 at 13:46 | comment | added | Joel David Hamkins | In the structure $\langle\mathbb{R},+,-,0,1,<\rangle$, without multiplication, one can mount an easy elimination-of-quantifiers argument that does not appeal to Tarski's far more substantial result on real-closed fields. Is this what you are asking? If so, I can explain it. | |
| Apr 24, 2017 at 13:23 | comment | added | Joel David Hamkins | In particular, without multiplication, I think things would be considerably easier. | |
| Apr 24, 2017 at 13:11 | comment | added | Joel David Hamkins | You wrote subtraction $-$, but did you mean multiplication? | |
| Apr 24, 2017 at 13:10 | answer | added | Joel David Hamkins | timeline score: 3 | |
| Apr 24, 2017 at 12:05 | comment | added | Mohammad Golshani | I am mainly interested in an argument like this: suppose $\mathbb{Z}$ is definable in the structure $\mathcal{R}$, by some formula and then work with the structure and the formula to get a contradiction. | |
| Apr 24, 2017 at 12:00 | history | edited | Mohammad Golshani | CC BY-SA 3.0 |
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| Apr 24, 2017 at 11:55 | history | edited | Mohammad Golshani | CC BY-SA 3.0 |
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| Apr 24, 2017 at 11:34 | answer | added | Mikhail Katz | timeline score: 6 | |
| Apr 24, 2017 at 11:25 | history | asked | Mohammad Golshani | CC BY-SA 3.0 |