Timeline for Can we have $\sf V=HTD$? How it relates to $\sf V=HOD$?
Current License: CC BY-SA 4.0
17 events
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| May 1, 2021 at 20:14 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| May 1, 2021 at 19:33 | comment | added | Zuhair Al-Johar | @JoelDavidHamkins, yes! That is nearer to the intention behind this method. | |
| May 1, 2021 at 19:31 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| May 1, 2021 at 17:21 | comment | added | Joel David Hamkins | You could alternatively (equivalently) say: V=HOD iff for every $x$ there is $\theta$ and $\alpha<\theta$ and $\varphi$ such that $x=\{a\in V_\theta\mid V_\theta\models\varphi(a,\alpha)\}$. | |
| May 1, 2021 at 17:12 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| May 1, 2021 at 17:06 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| May 1, 2021 at 16:42 | comment | added | Zuhair Al-Johar | @JoelDavidHamkins, thanks for the clarification, I'll write this in the posting in terms of types. | |
| May 1, 2021 at 16:39 | comment | added | Joel David Hamkins | In ZFC, the statement V=HOD can be expressed by saying: for every set x, there is an ordinal $\theta$ and an ordinal $\alpha<\theta$ and a formula $\varphi$ such that $x$ is the unique object in $V_\theta$ for which $V_\theta\models\varphi(x,\alpha)$. This does not use a truth predicate in $V$, but only in $V_\theta$, which we can prove exists in ZFC. The proof that this agrees with the external idea of definable in $V$ with ordinal parameters, however, is quite subtle. | |
| May 1, 2021 at 16:38 | comment | added | Zuhair Al-Johar | @JoelDavidHamkins, because I'm not experienced in this technical side of presenting ordinal definability. However, it appears to me that it could be done, anyhow I'll try read about his and try come up with the specific alternative that works. I was thinking naively of just replacing the ordinals with types in those definitions. | |
| May 1, 2021 at 16:28 | comment | added | Joel David Hamkins | OK, so why not make the definition that way then? Otherwise, one needs an account of your use of a truth predicate. | |
| May 1, 2021 at 16:20 | comment | added | Zuhair Al-Johar | @JoelDavidHamkins, yes but this alterantive can be done, because we have reflection here, and also we have something similar to the $V_\alpha$ sets which are the type sets. So, I think HOD can be paralled in types (not sure though). | |
| May 1, 2021 at 16:17 | comment | added | Joel David Hamkins | Your definition seems to use a truth predicate. This is not how we define HOD. It is a motivating idea, but that idea (because it uses a truth predicate) is not formalizable in ZFC. Rather, the definition of HOD employs reflection and the definability of the $V_\alpha$ sets. | |
| May 1, 2021 at 15:39 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| May 1, 2021 at 15:29 | comment | added | Zuhair Al-Johar | @Alex Kruckman, hmmm..., but this is so legitamate here that it's very hard to deviate from? It is at the heart of the matter! One needs to be cautious not to confuse both, the contexts are different. | |
| May 1, 2021 at 15:17 | comment | added | Alex Kruckman | "Type-definable" has a standard (and totally different) meaning in model theory, so I would recommend choosing different terminology. | |
| May 1, 2021 at 15:12 | history | edited | Asaf Karagila♦ |
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| May 1, 2021 at 14:51 | history | asked | Zuhair Al-Johar | CC BY-SA 4.0 |