Timeline for Is there a set $S\subseteq [0,1]$ with $|S|=2^{\aleph_0}$ and distinct pairwise distances?
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
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| Oct 12, 2018 at 15:40 | review | Close votes | |||
| Oct 16, 2018 at 15:56 | |||||
| Oct 12, 2018 at 11:55 | comment | added | Qfwfq | Don't delete the question please! As it turns out, it's a nice (routine?) application of Zorn's lemma: it has some didactical value. Maybe, just move it to MSE! | |
| Oct 12, 2018 at 10:46 | vote | accept | Dominic van der Zypen | ||
| Oct 12, 2018 at 10:27 | answer | added | KP Hart | timeline score: 8 | |
| Oct 12, 2018 at 10:14 | comment | added | Uri Bader | ... add to $R$ also all midpoints of pairs of elements in $S$... | |
| Oct 12, 2018 at 10:07 | comment | added | Uri Bader | Consider an infinite set $S$ with this property. Observe that the set $T=d(S,S)\subset [0,1]$ satisfies $|T|\leq |S|^2=|S|$. Consider the set $R=\{s\pm t\mid s\in S,~t\in T\}$. Observe that $|R|\leq 2\cdot |S|\cdot|T|= |S|$. Then $|S|<|[0,1]|$ implies $|R|<|[0,1]|$ and the set $S\cup \{x\}$ is good for every $x\in [0,1]-R$. | |
| Oct 12, 2018 at 10:02 | answer | added | dan_fulea | timeline score: 13 | |
| Oct 12, 2018 at 10:00 | comment | added | Dominic van der Zypen | Got it! I am convinced now every maximal element with dpdp is uncountable, but I'm not sure about $2^{\aleph_0}$ yet. As soon as I understand, I'll delete the question | |
| Oct 12, 2018 at 9:58 | comment | added | Uri Bader | We will catch it when it sneaks! If $S_\alpha$ is a chain with union $S$ and $x_1,x_2,y_1,y_2\in S$ having $d(x_1,y_1)=d(x_2,y_2)$ then they all should appear in some $S_\alpha$, don't they? | |
| Oct 12, 2018 at 9:52 | comment | added | Dominic van der Zypen | I don't see why a union of a chain of subsets with this property still has this property? Equal distances could "sneak in"... | |
| Oct 12, 2018 at 9:47 | comment | added | Uri Bader | I think it is hardly "research level"... | |
| Oct 12, 2018 at 9:46 | comment | added | Dominic van der Zypen | OK you are welcome to post it | |
| Oct 12, 2018 at 9:45 | comment | added | Uri Bader | but my answer remains correct: replacing "uncountable" with $2^{\aleph_0}$... | |
| Oct 12, 2018 at 9:43 | comment | added | Dominic van der Zypen | Sorry I forgot about non(CH) :-) | |
| Oct 12, 2018 at 9:42 | history | edited | Dominic van der Zypen | CC BY-SA 4.0 |
added 18 characters in body
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| Oct 12, 2018 at 9:42 | comment | added | Uri Bader | your edit was nasty... | |
| Oct 12, 2018 at 9:41 | history | edited | Dominic van der Zypen | CC BY-SA 4.0 |
added 18 characters in body
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| Oct 12, 2018 at 9:39 | comment | added | Uri Bader | yes, use Zorn to find a maximal such set and observe it is uncountable. | |
| Oct 12, 2018 at 9:33 | history | asked | Dominic van der Zypen | CC BY-SA 4.0 |