Timeline for Uncountability of the "Peculiar" sets:
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 19, 2011 at 13:10 | comment | added | Asaf Karagila♦ | I see your point now! Thank you very much, I guess these sets are more peculiar than I expected them to be... :-) | |
| Dec 19, 2011 at 0:39 | comment | added | Andreas Blass | @Asaf: Justin wrote that $A$ can't be partitioned into finite sets, and that proof looks OK to me. But I don't see how that leads to your claim that $A$ can't be partitioned into infinitely many infinite sets. Specifically, if I try to carry out a proof like Justin's in the case where the pieces are allowed to be infinite, there's no condition like Justin's $q$ that decides which $a_i$'s (with a ground-model set of indices $i$) constitute a particular piece of the partition. | |
| Dec 18, 2011 at 23:05 | comment | added | Asaf Karagila♦ | See Justin's answer in this link: mathoverflow.net/questions/12973/… In Cohen's basic model the D-finite set of reals is "strong" in the amorphous sense of the word (no partition into infinitely many non-singletons exists). | |
| Dec 18, 2011 at 4:27 | vote | accept | Somabha Mukherjee | ||
| Dec 17, 2011 at 22:38 | comment | added | Asaf Karagila♦ | @Andreas: While it does sound as a very convincing argument, I have run into several proofs that this set is a counterexample to the assertion that every set has a group operation, by the fact that every partition has only finitely many non-singletons. At this moment I am too tired to verify all the details of either arguments, and it will have to wait for tomorrow. | |
| Dec 17, 2011 at 22:25 | comment | added | Andreas Blass | @Asaf: If you view the elements of $A$ as subsets of $\omega$, won't there be infinitely many whose first element is $n$, for each $n\in\omega$? | |
| Dec 17, 2011 at 19:41 | comment | added | Asaf Karagila♦ | This set $A$ cannot even be split into infinitely many infinite sets, regardless to uncountability of either the partition or the parts. | |
| Dec 17, 2011 at 17:52 | history | answered | Andreas Blass | CC BY-SA 3.0 |