Timeline for Is Lebesgue/Borel non-measurability actually caused by non-uniqueness?
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jul 6, 2011 at 12:54 | comment | added | Goldstern | Thanks for pointing out that it was probably Sierpinski, not Gödel, who showed that a well-order of the reals in type omega1 yields a nonmeasurable set. | |
| Jul 6, 2011 at 12:53 | comment | added | Goldstern | Thank you, Andreas. I prefer the preorder that I gave over the (possibly more usual) well-order for two reasons: It is as canonical as the hierarchy of $L_\alpha$'s; well-ordering the countable differences $L_{\alpha+1} \setminus L_\alpha$ is less canonical, as you have to introduce some arbitrary order on the formulas (or Gödel operations). The second reason is related: In an exposition of $L$, this preorder can appear right after the definition of $L$, half a page before the well-order on $L$ is introduced. | |
| Jul 5, 2011 at 4:24 | comment | added | Andreas Blass | Martin's example is very close to the well-ordering suggested by Ali. The former is the pre-well-ordering obtained from the latter by obliterating the distinction between reals that are constructed simultaneously. Since the resulting equivalence classes are countable, Sierpinski's argument seems adequate for handling the pre-well-ordering as well as the well-ordering. | |
| Jul 4, 2011 at 17:29 | comment | added | Ali Enayat | @Martin: as an example of a $\Delta^1_2$ non-measurable set in $ZF+V=L$ isn't it easier to look at the well-ordering $W$ of the reals of order-type $\aleph_1$ in $L$? Sierpinski had already observed, using Fubini's theorem, that no such $W$ can be measurable. | |
| Jul 4, 2011 at 13:03 | history | answered | Goldstern | CC BY-SA 3.0 |