Timeline for The smallest initial ordinal which is not defined using first-order formulas with parameters of smaller ordinals?
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Oct 13, 2017 at 8:51 | comment | added | Yair Hayut | A remark about the title of the question: In Dor Marciano's master thesis, he analyzed the consistency strength of having the first undefinable ordinal ($\theta_0$) an uncountable cardinal (which is roughly inaccessible) or inaccessible (which is roughly Mahlo) and so on. He has a general technique for controlling this ordinal, which you might find interesting. | |
| Oct 12, 2017 at 3:08 | vote | accept | Zetapology | ||
| Oct 12, 2017 at 3:07 | vote | accept | Zetapology | ||
| Oct 12, 2017 at 3:08 | |||||
| Oct 12, 2017 at 2:33 | answer | added | Noah Schweber | timeline score: 2 | |
| Oct 11, 2017 at 22:46 | history | edited | Zetapology | CC BY-SA 3.0 |
Corrected mistake
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| Oct 11, 2017 at 22:45 | comment | added | Zetapology | It would also be sufficient to simply say "In ZFC + $\Theta_0$ exists, it is _____" where _____ is replaced with some large cardinal property. In fact, I would even be satisfied with $\Theta_0$'s cofinality. | |
| Oct 11, 2017 at 21:30 | review | Close votes | |||
| Oct 12, 2017 at 23:46 | |||||
| Oct 11, 2017 at 17:03 | comment | added | Zetapology | I would simply like to see if any models of ZFC have $\theta_0$ and if so what axioms are required to restrict ZFC to said models. | |
| Oct 11, 2017 at 16:27 | comment | added | Miha Habič | I feel like this is quite like your previous question mathoverflow.net/q/281377/1058. As was explained there, by Yair, it is not clear without further assumptions that even $\theta_0$ exists, so could you spell out exactly what you have in mind? Do you want your models to have access to a truth predicate, for example? | |
| Oct 11, 2017 at 14:30 | comment | added | Zetapology | @Yair I restrict $\phi$ to be finite and first-order. I also restrict you to only look at "Tall" models (models which contain every ordinal in $V$) | |
| Oct 11, 2017 at 13:29 | comment | added | Noah Schweber | Following up on Yair's comment: arxiv.org/abs/1105.4597 | |
| Oct 11, 2017 at 8:52 | comment | added | Yair Hayut | Do you restrict the complexity of $\phi$ in some way? Otherwise, it is possible that all ordinals are definable without parameters. | |
| Oct 11, 2017 at 4:51 | history | edited | Zetapology | CC BY-SA 3.0 |
Fixed grammar
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| Oct 11, 2017 at 4:32 | history | asked | Zetapology | CC BY-SA 3.0 |