Timeline for When do we have a bijection between a proper class A and its power set class P(A)?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Dec 6, 2014 at 15:40 | comment | added | Asaf Karagila♦ | @Gerard: I should have been more careful. I read the question as asking about "Every class $W$ has a bijection with $\mathcal P(W)$", which is what my answer shows. | |
| Dec 6, 2014 at 15:39 | comment | added | Gérard Lang | Dear Asaf, I read your exchanges with Sam and understand that he saw that your proof was not sufficient. As I did not see that, please accept my excuses that I accepted your answer to my question as correct !. GL | |
| Dec 6, 2014 at 15:08 | comment | added | Asaf Karagila♦ | @Gerard: As Sam points out, I misread the question. Please unaccept my answer. I will leave it for a while, if only to keep the comments below it visible. | |
| Dec 6, 2014 at 13:49 | comment | added | Sam Roberts | @AsafKaragila Sorry to be dense, but how does the general claim (for all $A$, if $A$ is bijective with $\mathcal P(A)$, then $A$ is bijective with $V$) follow from the instance $A = On$ (if $On$ is bijective with $\mathcal P(On)$, then $On$ is bijective with $V$)? | |
| Dec 6, 2014 at 11:28 | comment | added | Asaf Karagila♦ | You just need to notice that $\mathcal P\sf (Ord)$ can always be linearly ordered. In a model where there is a class which cannot be linearly ordered, this class cannot be put into bijection with $\mathcal P\sf (Ord)$; and if this class is such that there is no injection from $\sf Ord$ into it, then this class cannot be put into bijection with $V$ either. Joel Hamkins suggested such model in an answer to a question of mine, to which I linked in one of your recent questions. | |
| Dec 6, 2014 at 11:08 | vote | accept | Gérard Lang | ||
| Dec 6, 2014 at 15:58 | |||||
| Dec 6, 2014 at 11:08 | comment | added | Gérard Lang | Dear Asaf, Thank you very much for this nice answer. If I correctly understand, the negation of the axiom of global choice is equivalent with the existence of a proper class A such that A is bijective with P(A) and not bijective with V. And by the model built by Ali Enayat, where On, P(On) and P(P(On)) are not bijective, it is possible that such A is neither bijective with On, nor with P(On). Would it be possible that the class W defined by JD Hamkins be such a class ? | |
| Dec 6, 2014 at 9:00 | history | answered | Asaf Karagila♦ | CC BY-SA 3.0 |