Timeline for Construction of nonmeasurable sets
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Aug 29, 2014 at 23:08 | comment | added | Asaf Karagila♦ | @Monroe: First of all, how? Secondly, assume $\sf ZF+2^{\aleph_0}=\aleph_1$. How would you prove such measure space cannot exist in this theory? So the same way would work for the atoms. Recall that in many of the badly behaved permutation models, there are very few and usually quite trivial maps between the "permutation generating objects" and pure sets. (Thirdly, don't forget to ping me if you want me to see your comments! You can always send me an email if you want to elaborate, by the way.) | |
| Aug 29, 2014 at 22:05 | comment | added | Monroe Eskew | So how does that block the possibility of a measure space with a total measure in the FM model? Because it would induce a total measure on $\mathbb{R}$? | |
| Aug 29, 2014 at 21:51 | comment | added | Asaf Karagila♦ | @Monroe: Given a measure space in a permutation model, we can transfer it, in full, to a symmetric extension using the Jech-Sochor theorem, and you can even arrange that $\Bbb R$ is well-ordered in that symmetric extension. So atoms really don't change the picture here. | |
| Aug 29, 2014 at 21:45 | comment | added | Monroe Eskew | @Asaf: Maybe one could still investigate measurability in these models with respect to more general (outer-) measure spaces involving the urelements? | |
| Aug 29, 2014 at 21:08 | comment | added | Asaf Karagila♦ | @Monroe: Nothing to investigate. Since the reals are considered as pure sets. In permutation models pure sets are well-ordered. So everything works as in $\sf ZFC$. | |
| Aug 29, 2014 at 20:23 | comment | added | Monroe Eskew | Now I am very curious whether anyone investigated measure theory in Fraenkel-Mostowski permutation models. | |
| Aug 29, 2014 at 20:02 | comment | added | Monroe Eskew | And Halpern did it without forcing! | |
| Aug 29, 2014 at 19:49 | comment | added | Monroe Eskew | This is a very good answer. It shows that there was some work on finding nonmeasurable sets without the full AC, perhaps not by outright trying to prove it with only DC, but by weakening the hypothesis. | |
| Aug 29, 2014 at 16:51 | history | answered | godelian | CC BY-SA 3.0 |