Timeline for Why the restrictions in the definition of Berkeley cardinals?
Current License: CC BY-SA 4.0
11 events
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| Jul 20, 2019 at 16:48 | comment | added | Tim Campion | Unless I'm wildly off-base -- when we say "elementary embedding from $M$ to $M$" we mean "elementary embedding from $(M,\in)$ to $(M,\in)$", right? Another issue is it seems to me that if $M$ is not transitive, then a nontrivial elementary embedding $(M,\in) \to (M,\in)$ might not move any ordinals... It seems like the Mostowski collapse would be more relevant to the question of generalizing the Berkeley condition to $(M,R)$ where $R \neq \in$... | |
| Jul 20, 2019 at 16:44 | comment | added | Tim Campion | @NoahSchweber I'm confused. As I understand, the Mostowski collapse is used to take a set $X$ with a well-founded, extensional binary relation $R$, and build an isomorphic transitive set $\hat X$ such that $(X,R)$ is isomorphic to $(\hat X,\in)$. I think you're saying to take $(X,R) = (M,\in)$ and Mostowski collapse. But if $M$ is not transitive, then $(M,\in)$ need not be extensional -- e.g. take $M = \kappa^+ \cup \{\{\{1\}\}\}$, where $\emptyset \in M$ and $\{\{1\}\} \in M$ have the same elements in $M$. So the Mostowski collapse of $M$ in the sense I understand doesn't exist... | |
| Jan 3, 2019 at 21:45 | vote | accept | Zuhair Al-Johar | ||
| Jan 2, 2019 at 19:38 | comment | added | Noah Schweber | @ZuhairAl-Johar Yes, but I don't think efficiency is always the highest goal. Focusing explicitly on transitive sets makes the picture easier to think about, in my opinion; we don't have to pay attention to Mostowski collapses. It also plays more nicely with other notions where transitivity/$\ni\kappa$ is more important. Re: the class condition, like I said (in my edit) I don't know. | |
| Jan 2, 2019 at 19:36 | comment | added | Zuhair Al-Johar | it is much shorter to say $\kappa \subseteq M$ than saying $M$ is transitive and $\kappa \in M$. What remain is the class condition? | |
| Jan 2, 2019 at 19:10 | history | edited | Noah Schweber | CC BY-SA 4.0 |
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| Jan 2, 2019 at 19:10 | comment | added | Noah Schweber | @Wojowu Yeah I noticed that just as I finished typing :P. Fixed! | |
| Jan 2, 2019 at 19:09 | comment | added | Wojowu | You quoted Kunen's inconsistency theorem. | |
| Jan 2, 2019 at 19:09 | comment | added | Wojowu | The question is definitely in the context of ZF, not ZFC. In ZFC we already have a problem by taking $M=V_{\kappa+2}$. | |
| Jan 2, 2019 at 19:03 | history | edited | Noah Schweber | CC BY-SA 4.0 |
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| Jan 2, 2019 at 18:55 | history | answered | Noah Schweber | CC BY-SA 4.0 |