Timeline for What is the limit to iterating class comprehension, reflection and limitation of size?
Current License: CC BY-SA 4.0
17 events
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| Jul 15, 2019 at 18:58 | history | edited | Master | CC BY-SA 4.0 |
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| Jul 6, 2019 at 19:43 | comment | added | Zuhair Al-Johar | I see your point. Thanks! | |
| Jul 6, 2019 at 19:41 | comment | added | Master | I know. But the point is that there is a distinction between definable and true variants of properties. If you want $Ord$ to be actually Mahlo, you'll have to add it as a new axiom. | |
| Jul 6, 2019 at 19:32 | comment | added | Zuhair Al-Johar | but limitation of size is already an axiom of this theory. | |
| Jul 6, 2019 at 17:18 | comment | added | Master | They didn't prove $V^1$ is Mahlo, only that it is definably Mahlo. You can prove $V^1$ is inaccessible only be assuming limitation of size, which is much stronger then Replacement. | |
| Jul 6, 2019 at 13:59 | vote | accept | Zuhair Al-Johar | ||
| Jul 6, 2019 at 6:18 | comment | added | Zuhair Al-Johar | so according to that $Ord^2$ which is the class of all ordinals in $V^2$ is not provably a Mahlo in $T$, I just thought that the size limitation axioms coupled with class comprehension axioms would ensure that this must be the case, as they did it with the prior universe $V^1$ which has a similar situation with replacement, there they managed to prove that $V^1$ is inaccessible, and thus $T \vdash ZFC$, so I thought a similar thing would also apply for the case of $V^2$ such that $T \vdash Ord^2 is Mahlo$. Why that failed at this level? | |
| Jul 5, 2019 at 18:35 | comment | added | Master | ($T$ is your theory) | |
| Jul 5, 2019 at 18:30 | comment | added | Master | "The intended models are $V_\kappa$, where $\kappa$ is Mahlo." There a plenty of models of "$ORD$ is Mahlo" which are not of the form $V_\kappa$, where $\kappa$ is Mahlo be elementaryity (Take $V_\lambda\prec V_\kappa$, where $\kappa$ is the least Mahlo). But that is besides the point. Sure $V^2$ is a model of "$ORD$ is Mahlo," but $T\nvdash (V^2\vDash ORD\,is\,Mahlo)$. It does prove every individual instance of that schema. On the other hand, "$ORD$ is Mahlo" proves a proper class of $V_\kappa$, where $\kappa$ is reflecting, and each $V_\kappa$ satisfies the condition for $V^\lambda$. | |
| Jul 5, 2019 at 18:15 | comment | added | Zuhair Al-Johar | I see what you mean. I myself made a mistake in my comment, I meant $V^2$ when I said $V^1$. So again I thought that Packomov's answer established that $V^2$ is a model of $ZFC+M$, this means that $V^2=V_\kappa$ where $\kappa$ is a Mahlo, so the set of all ordinals in $V^2$ is a Mahlo cardinal, and so it proves the consistency of $ORD$ is a Mahlo, so it is already stronger than $ORD$ is a Mahlo, so as I said we are already way beyond that. So there must be something wrong with your argument? | |
| Jul 5, 2019 at 16:29 | history | edited | Master | CC BY-SA 4.0 |
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| Jul 5, 2019 at 16:20 | comment | added | Master | It is a schema. If $C=\{\alpha|\phi(\alpha,p)\}$ is club, there there is some regular $\kappa\in C$. It is not a single assertion. Here is a link to Cantors attic: cantorsattic.info/ORD_is_Mahlo | |
| Jul 5, 2019 at 16:18 | comment | added | Zuhair Al-Johar | Can you explicitly write $ORD$ is mahlo formally | |
| Jul 5, 2019 at 15:49 | comment | added | Master | I believe the reason is that, while $V_\kappa$ might satisfy any individual axiom of "$ORD$ is Mahlo," you can't prove it satisfies all of them. | |
| Jul 5, 2019 at 9:30 | comment | added | Zuhair Al-Johar | I just was under the impression that $K^+(V^2)$ already interprets "$Ord $ is Mahlo", I mean the second tier of this theory already proves the consistency of $Ord$ is Mahlo. The reason is because the proof in the linked answer already establishes $V^1$ as a a model of $\sf ZF + M$. Where $\sf M$ is the schema presented in the linked answer. So there exists $\kappa$ where $V^1=V_\kappa$ where $\kappa$ is a Mahlo. So I thought that we are already beyond Mahlo cardinals. | |
| Jul 5, 2019 at 4:21 | history | edited | Master | CC BY-SA 4.0 |
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| Jul 5, 2019 at 0:31 | history | answered | Master | CC BY-SA 4.0 |