Timeline for Can proper classes have different sizes?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Mar 13 at 11:22 | comment | added | Ali Enayat | Also related are the following: mathoverflow.net/questions/124494/… AND mathoverflow.net/questions/280526/… | |
| Mar 13 at 6:44 | comment | added | Joel David Hamkins | @Anonymousgradstudent Since every class is a subclass of $V$, this will necessarily be the largest class. | |
| Mar 13 at 6:44 | comment | added | Joel David Hamkins | @MikeBattaglia The axiom V=HOD is equivalent over ZFC to the existence of a first-order definable (without parameters) global well-ordering of the universe. | |
| Mar 13 at 6:23 | comment | added | Anonymous grad student | Thanks for the tip about upvoting :). And thanks for confirming re: global choice! A small followup, given the extremely helpful last reply: are there any well-studied set theories in which proper classes have different sizes, but the equinumerosity class of V is not the largest one? | |
| Mar 13 at 1:12 | comment | added | Mike Battaglia | @JoelDavidHamkins is the axiom V = HOD related to this somehow? I thought I'd read, somewhere, that V = HOD was somehow related to global choice in this kind of way. | |
| Mar 12 at 23:58 | comment | added | Joel David Hamkins | Yes, that's right. If you want classes to exhibit different sizes, you need to drop global choice, but then it is possible. There will be a largest size, the equinumerosity class of $V$, and other sizes, such as the equinumerosity class of Ord. The class of surreal numbers is equinumerous with the class of all ordinal binary sequences $2^{<\text{Ord}}$. (By the way, if you find a post helpful, you can vote it up by clicking the uparrow button.) | |
| Mar 12 at 23:52 | comment | added | Anonymous grad student | This is, of course, incredibly helpful. So (just to restate one of your points, to make sure I'm tracking): if I were working in GBc, and I wanted to think about proper classes w different cardinalities, then I would just drop the axiom of global choice and think about how to understand the different sizes of proper classes in that context? | |
| Mar 12 at 23:44 | history | answered | Joel David Hamkins | CC BY-SA 4.0 |