Timeline for What is $\omega_1^{CK}(\mathsf{Ord})$?
Current License: CC BY-SA 4.0
21 events
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| Jun 9 at 4:37 | history | edited | Noah Schweber | CC BY-SA 4.0 |
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| Dec 12, 2019 at 22:05 | comment | added | Fedor Pakhomov | Yes, right, I have misinterpreted Andreas. | |
| Dec 12, 2019 at 21:31 | comment | added | Noah Schweber | @FedorPakhomov Coming back to your original comment, I think Andreas' definition does in fact work: he's looking at the supremum of all ordertypes of definable genuine well-orderings. So I don't think there's too much of a problem here. | |
| Dec 10, 2019 at 20:46 | comment | added | Noah Schweber | @ElliotGlazer I wasn't aware of that, thanks! | |
| Dec 10, 2019 at 20:44 | comment | added | Elliot Glazer | There's also Joel's answer here mathoverflow.net/a/270540/109573 | |
| Dec 10, 2019 at 20:05 | comment | added | Noah Schweber | @FedorPakhomov I see, thank you very much! I'll read the relevant parts of Girard's book and post any more questions I have to MO/MSE proper. | |
| Dec 10, 2019 at 19:51 | comment | added | Fedor Pakhomov | Prae-dilator is an endo-functor acting on the category of linear orders that preserves directed co-limits and pullbacks. (Dilators are the same but for the category of well-orders). | |
| Dec 10, 2019 at 19:48 | comment | added | Noah Schweber | @FedorPakhomov Thank you very much, that's quite interesting! Quick question - what does "prae" mean? I've not seen that notation before. And I wasn't aware of Girard's book, that's a great resource. | |
| Dec 10, 2019 at 19:46 | comment | added | Fedor Pakhomov | With regard to the references. I read about $\beta$-completeness theorem in the draft of Girard's unpublished book girard.perso.math.cnrs.fr/Archives4.html (Chapter 10). Maybe there is a better reference, but I don't know about that. $F$ in the argument above is the dilator corresponding to the canonical $\beta$-proof of the sequent $\mathsf{ZFC}'\Rightarrow$. Theory $\mathsf{ZFC}'$ here is an alternative presentation of $\mathsf{ZFC}$ given in a two sorted language with a separate sort for ordinals. | |
| Dec 10, 2019 at 19:31 | comment | added | Fedor Pakhomov | I meant that the theory of ptykes is too complicated to outline. Actually now I figured out that this could be done just using dilators; I outline the argument assuming some familiarity with the theory. Using Girard's completeness theorem for $\beta$-logic it is possible to construct a prae-dilator $F$ s.t. $F(\alpha)$ is w.o. iff there are no transitive models of $\mathsf{ZFC}$ of the height $\le \alpha$. Notice that $F$ is a dilator in the least transitive model $M\models\mathsf{ZFC}$. Thus $M\models WO(F(On))$. However, externally $F(On^M)=(F(On))^M$ is ill-founded. | |
| Dec 10, 2019 at 18:51 | comment | added | Noah Schweber | @FedorPakhomov Is there a citation for that construction? I'd be really interested in reading that (I've been vaguely interested in ptykes for a while, motivated by van de Wiele's theorem). | |
| Dec 10, 2019 at 18:50 | comment | added | Noah Schweber | @FedorPakhomov But $L_{\omega_1^{CK}}\not\models$ ZFC, and indeed the missing piece is that KP$\omega$ can't prove that every set-well-ordering is in bijection with an ordinal. | |
| Dec 10, 2019 at 18:50 | comment | added | Fedor Pakhomov | I do know a construction of a transitive model of $\mathsf{ZFC}$ with internally well-founded, but externally ill-founded class. However it uses theory of ptykes (higher-order operators on well-orderings) and there is just too much to explain about that. | |
| Dec 10, 2019 at 18:49 | comment | added | Fedor Pakhomov | There are models that are wrong about some class being well-ordered.The most well-known example would be that $L_{\omega_1^{CK}}$ thinks that Gandy order (ill-founded order without hyperarithmetic descending chains) is well-founded. Although it isn't particular interesting since Gandy order is set-size and set-size order couldn't be externally ill-founded for models of theories that unlike KP could prove that any well-ordering is isomorphic to an ordinal. | |
| Dec 10, 2019 at 18:34 | history | edited | Noah Schweber | CC BY-SA 4.0 |
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| Dec 10, 2019 at 18:33 | comment | added | Noah Schweber | @FedorPakhomov Oof, I didn't know that! I've changed the notation. As to your initial comment, can that happen? At a glance I don't see how to do it (the obstacle being that ZFC proves that every set-well-ordering is isomorphic to some ordinal, and $M$ is transitive; of course the candidate bad orders can be horribly non-set-like in $M$ this isn't fatal, but it does slow me down). | |
| Dec 10, 2019 at 18:28 | comment | added | Fedor Pakhomov | I would suggest to rename $\mathsf{KPi}$ to $\mathsf{KP}\omega$. At least in proof-theory people usually denote $\mathsf{KPi}$ the theory $\mathsf{KP}+\text{every set lies in a transitive model of }\mathsf{KP}$ (i here stands for (recursively) inaccessible). | |
| Dec 10, 2019 at 18:27 | history | edited | Noah Schweber | CC BY-SA 4.0 |
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| Dec 10, 2019 at 18:25 | comment | added | Fedor Pakhomov | I think that in general (not assuming $M^\omega\subseteq M$) there are problems with the definition that Andreas Blass has gave. Namely, $M$ might think that some class is well-ordered, although from external point of view it isn't well-ordered (which would make $\omega_1^{CK}(M)$ ill-founded). So the natural condition here is to only consider $M$ such that $M\models L\text{ is a class well-ordering}$ implies $WO(L^M)$. But for $M$ of this sort the property to be $M$-definable well-ordering will be $\Delta_0$ in $Ad(M)$. And your argument will work for them in the same manner. | |
| Dec 10, 2019 at 18:18 | history | edited | Noah Schweber | CC BY-SA 4.0 |
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| Dec 10, 2019 at 18:02 | history | answered | Noah Schweber | CC BY-SA 4.0 |