Timeline for An axiom for collecting proper classes
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 8, 2018 at 14:34 | comment | added | Julia Williams | @Alec [email protected] | |
| Feb 8, 2018 at 14:01 | comment | added | Alec Rhea | @KamerynWilliams Thank you for the insight! I can't seem to find an email address to contact you at privately, but mine is [email protected] -- please feel free to mail me there or post your email here and I will follow up, I would thoroughly enjoy looking at the chapter in question from your upcoming paper. | |
| Feb 7, 2018 at 21:28 | comment | added | Julia Williams | The chapter Joel mentioned isn't quite to the point of being publicly shared around, but I can post a link to it within a few months. Alternatively, feel free to contact me privately. In brief, while ETR is enough to get that the unrolling satisfies the basic axioms of set theory, the resulting structure has a fairly weak theory. Most likely, you just want to work with KMCC, and not sweat the small difference in consistency strength. | |
| Feb 7, 2018 at 20:54 | comment | added | Julia Williams | You can pick representatives from 'full-sized' hyperclasses under some extra assumptions. If your unrolled structure is of the form $L[A]$—which, by work of Antos and Friedman (arxiv.org/abs/1510.04082), can be forced over MK + Class Collection + the Ord-Dependent Choice schema—then you have a definable global well-order of the unrolled structure, which you can use to make the choices. And the MK + ... theory is equiconsistent with MK, so these extra assumptions aren't costly. | |
| Feb 7, 2018 at 20:51 | comment | added | Joel David Hamkins | You can find a little about ETR in my paper on class games with Victoria Gitman, and also in my work with several co-authors on the class forcing theorem. The work on KMCC is not yet available. Kameryn's dissertation results will be available in a few months if not sooner. | |
| Feb 7, 2018 at 20:22 | comment | added | Alec Rhea | @JoelDavidHamkins Thank you! My foundational needs are settled once again using ETR -- I would very much enjoy reading more about the set-theoretic strength required for an 'unrolled' version of the universe where this algebra takes place legitimately. I would also enjoy reading more about the class collection axiom if there is a freely available link to your work with Victoria Gitman (I can't seem to find it on google). | |
| Feb 7, 2018 at 20:08 | vote | accept | Alec Rhea | ||
| Feb 7, 2018 at 19:57 | comment | added | Joel David Hamkins | Yes, I agree completely. In that case, one wants to be working in a ZFC set theory with universes of some kind, such as a proper class of inaccessible cardinals, or possibly Ackermann set theory. | |
| Feb 7, 2018 at 19:48 | comment | added | Andreas Blass | This seems like the best solution if the desired hyperclasses are only "class-sized" (analogously to the "hereditarily countable" requirement for unrolling as reals). So it should solve the OP's difficulties. If one wants "full-sized" hyperclasses, then MK won't suffice. | |
| Feb 7, 2018 at 19:28 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |