Timeline for Axiom of choice and the equality between second-order constructible universe and HOD
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Nov 29, 2022 at 22:46 | history | edited | Hanul Jeon | CC BY-SA 4.0 |
added 6 characters in body
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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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| Feb 20, 2015 at 15:33 | vote | accept | Hanul Jeon | ||
| Feb 20, 2015 at 13:14 | answer | added | Mohammad Golshani | timeline score: 8 | |
| Feb 19, 2015 at 15:27 | comment | added | Asaf Karagila♦ | (Of course the existence of elementary submodels of this type generally requires $\sf DC$ and $\sf AC_\kappa$ for every aleph number $\kappa$.) | |
| Feb 19, 2015 at 15:25 | comment | added | Asaf Karagila♦ | @Joel: That much is clear, but the interesting question is whether or not the proof can be modified slightly to accommodate this. For example, if you could ensure that instead of $V_\alpha$, you could take some elementary submodel of $V_\alpha$ which can be encoded by ordinals and that it calculates $A$ correctly. | |
| Feb 19, 2015 at 15:23 | comment | added | Joel David Hamkins | The proof is definitely using AC in a fundamental way, since when a set is in HOD, then the proof argues that it is definable in some $V_\theta$, and then you wait for the $L_{SO}$ hierarchy to grow large enough that you can have a predicate coding all of $V_\theta$ in order to define $x$ at that stage. But if $V_\theta$ is not well-orderable, there can be no such stage where $V_\theta$ is coded like that. | |
| Feb 19, 2015 at 15:10 | comment | added | Asaf Karagila♦ | Earlier this year one of the talks in the Friday set theory seminar was (amongst other things) about this construction, and later we had a discussion one whether or not the axiom of choice is used there. I can't recall the answer or semi-answer that we arrived at, though. | |
| Feb 19, 2015 at 14:18 | review | First posts | |||
| Feb 19, 2015 at 14:30 | |||||
| Feb 19, 2015 at 14:16 | history | asked | Hanul Jeon | CC BY-SA 3.0 |