Timeline for Critical points and the Foundation Axiom
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 1, 2016 at 0:07 | comment | added | Thomas Benjamin | Considering that my question has been downvoted, I am considering deleting the question. Since I have accepted your answer, I am not sure whether I should. Please advise. Thanks. | |
| Jun 30, 2016 at 10:58 | comment | added | Thomas Benjamin | ($WF$ is a transitive class with respect to $\in$) | |
| Jun 30, 2016 at 10:38 | comment | added | Thomas Benjamin | (cont.) really classes of well-ordered sets, and the Kunen inconsistency holds of course for these. Since for $<$$WF$,$\in$$>$, $\in$ is rigid for $WF$ (since $WF$ is a transitive class), I conjectured that there are no nontrivial elementary embeddings $j$:$WF$$\rightarrow$$WF$ and asked essentially whether the results of Theorem 1 of the paper in question can, in fact be extended to the class $WF$ (a closer look at the theorem suggests--probably not....). | |
| Jun 30, 2016 at 9:49 | comment | added | Thomas Benjamin | Since $V_{ZF}$=$WF$ and $V_{ZFC}$=$WO$ ($WO$=the class of all well-ordered sets), the $V$ for $ZFC^{-f}$ is still a class of well-orderd sets (since Choice still holds for $ZFC^{-f}$). Recall that in the paper mentioned, the theorem ("Work in $GBC^{-f}$ or $ZFC^{-f}$. Then there is no nontrivial $\Sigma_1$-elementary embedding $j$:$WF$$\rightarrow$$WF$.") was billed as stating that the Kunen inconsistency still holds for the class $WF$ of $GBC^{-f}$ or $ZFC^{-f}$. Though true, it seems to me a bit of 'false advertising', since the '$WF$' for $GBC^{-f}$ and $ZFC^{-f}$ are | |
| Jun 30, 2016 at 5:07 | comment | added | Noah Schweber | @ThomasBenjamin Yes, but I still don't see the connection - how does the fact that "$\bigcup V_\alpha=V$ is equivalent to foundation over ZF" have anything to do with extending the result mentioned to ZF? | |
| Jun 30, 2016 at 1:07 | comment | added | Thomas Benjamin | Regarding the second point in your answer--this is because $WF^{V}$ (since $WF^{V}$ is a model of $ZFC^{-f}$) is a class of well-ordered sets (since Choice still holds for $WF^{V}$)? | |
| Jun 30, 2016 at 1:00 | vote | accept | Thomas Benjamin | ||
| Jun 29, 2016 at 19:11 | history | answered | Noah Schweber | CC BY-SA 3.0 |