Timeline for Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem)
Current License: CC BY-SA 3.0
16 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
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| S Feb 19, 2015 at 14:23 | history | suggested | Hanul Jeon | CC BY-SA 3.0 |
I think the inclusion is contextually correct.
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| Feb 19, 2015 at 14:06 | review | Suggested edits | |||
| S Feb 19, 2015 at 14:23 | |||||
| Apr 16, 2014 at 15:05 | comment | added | Thomas Benjamin | @Prof. Hamkins: Can V as characterized in your theorem contain sets nonconstructible relative to L? Do AC and GCH hold in V as characterized by your theorem? In your conception of the set-theoretic multiverse, is there a forcing extension M[G] of this V? | |
| Feb 7, 2014 at 18:53 | comment | added | user46667 | Great! Chang's paper is a very nice reference that exactly matches my questions. Thank you very much. I don't know if Chang (or others) investigated around possible relations between these generalized constructible universes and Kunen inconsistency and HOD problem or not. | |
| Feb 7, 2014 at 18:50 | vote | accept | CommunityBot | ||
| Feb 7, 2014 at 18:32 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
Myhill Scott reference
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| Feb 7, 2014 at 18:22 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
Chang model
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| Feb 7, 2014 at 18:21 | comment | added | user46667 | In order to give a proof for Kunen's inconsistency theorem using non-first order logics, maybe using the power of logics $\mathcal{L}_{\infty,\infty}$ and $SO$ one can express some expressions which produce a contradiction when we assume there exists a non-trivial elementary embedding from $V$ to itself (and similarly from $HOD$ to itself). | |
| Feb 7, 2014 at 18:00 | comment | added | Noah Schweber | Oh, right. That makes more sense. | |
| Feb 7, 2014 at 17:53 | comment | added | Joel David Hamkins | @NoahS, since $V_\kappa$ could be much larger than $\kappa$ in size, I think $L([\kappa]^{<\kappa})$ is closer to being right. I'd have to think more about it. | |
| Feb 7, 2014 at 17:46 | comment | added | Noah Schweber | Joel, it seems your argument also shows that $L_\kappa$ (in Gina's sense) is $L(V_\kappa)$; is this correct? | |
| Feb 7, 2014 at 17:26 | comment | added | user46667 | Is it possible to find a proof of Kunen's inconsistency theorem in terms of infinitary logic by analyzing the structure of $L_{\infty}$ ($=V$)? If yes, is it possible to reformulate this proof for similar result about $L_{SO}$ ($=HOD$)? Maybe there is no non-trivial self-elementary embedding for $V$ because $V$ is $L$ of a too powerful logic like $\mathcal{L}_{\infty,\infty}$ as well as $HOD$ which is $L$ of another too powerful logic like $SO$. | |
| Feb 7, 2014 at 17:15 | comment | added | user46667 | It is a really interesting observation that in this dual approach for building Godel's costructible universe by strengthening the expression power of our logic using longer formulas we will reach $V$ and if we increase the expression power of the logic by assuming wider range for quantifiers we will reach $HOD$. It seems $HOD$ and $V$ are some kind of limit $L$ in limit logics (The first one in $\mathcal{L}_{\infty,\infty}$ and the next one in $SO$). Maybe this means one can prove a dual form of Kunen's inconsistency theorem for $HOD$ too. | |
| Feb 7, 2014 at 14:04 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
added 861 characters in body
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| Feb 7, 2014 at 13:48 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |