Timeline for What are some kinds of models where DC holds?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Jul 29, 2017 at 13:54 | comment | added | Asaf Karagila♦ | I bet your forgot all about thinking about this! :) | |
| Jul 8, 2017 at 18:12 | comment | added | Asaf Karagila♦ | Ah, of course. I only remembered the $\Delta_5$ symbol, and somehow in my mind it didn't represent the "class of DF" but rather $\Delta^0_5$ in the Borel hierarchy. :-) | |
| Jul 8, 2017 at 17:50 | comment | added | Noah Schweber | @AsafKaragila Sorry for the late reply. Miller's paper shows that the DF set is $F_{\sigma\delta}$, if I recall correctly. The point about codes is well taken - I'll need to think about it (and it might break this answer). | |
| Jun 29, 2017 at 15:03 | comment | added | Asaf Karagila♦ | Also, how did you figure that the DF set is $F_{\sigma\delta}$? | |
| Jun 28, 2017 at 4:58 | comment | added | Asaf Karagila♦ | If I recall correctly, "Every Borel set has a code" is equivalent to countable choice for sets of reals. Look at the Feferman--Levy model where the real numbers are a countable union of countable sets. There every set is $F_{\sigma\sigma}$, but not every set has a code. I think that's the problem here. | |
| Jun 27, 2017 at 19:31 | comment | added | Asaf Karagila♦ | Well, you might be right. I'll have to look at this again tomorrow morning. I don't have a mind set for careful details right now... :) | |
| Jun 27, 2017 at 19:29 | comment | added | Noah Schweber | @AsafKaragila Well, is there any flaw in the argument that infinitude and Dedekind-finitude of the Borel set coded by $r$ transfers from $V$ to $L(\mathbb{R})^V$, whenever $V\models ZF$? Because that's all I need here, and it seems solid ... | |
| Jun 27, 2017 at 19:16 | comment | added | Asaf Karagila♦ | What worries me is that Feferman's model without ultrafilters on $\omega$ has the same sets of ordinals and therefore the same reals as the Cohen model. And I'm not entirely sure that DC fails there. | |
| Jun 27, 2017 at 19:15 | comment | added | Noah Schweber | @AsafKaragila My point is that if $V$ contains a Borel set which it thinks is infinite Dedekind-finite, then so does $L(\mathbb{R})^V$. So we look at $L(\mathbb{R})$ of the Cohen model. | |
| Jun 27, 2017 at 18:47 | comment | added | Asaf Karagila♦ | Are you thinking about Cohen's first model? Because it is not of the format $L(\Bbb R)$... | |
| Jun 27, 2017 at 16:08 | history | answered | Noah Schweber | CC BY-SA 3.0 |