Timeline for Large cardinal near inconsistencies
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 27, 2023 at 19:02 | comment | added | Joseph Van Name | @C7X I have seen plenty of evidence (from computer experiments) for the consistency of I0-I3 (and much more). I am not at all concerned about any mathematics below I0 that is lost due to any inconsistency. | |
| Jun 26, 2023 at 22:00 | comment | added | C7X | @JosephVanName Incidentally Dowd's recent preprint "Mahlo Rank and Mitchell Order" contains a claimed proof of inconsistency of ZFC+Supercompact, if so much work on I0-I3 may be ruled out. | |
| Jun 26, 2023 at 21:01 | comment | added | Holo | @JosephVanName considering that Berkeley cardinals are known to be inconsistent (with AC), the assertion "If Berkeley cardinals are found to be inconsistent, not much would change" is as true as you can get | |
| Jun 26, 2023 at 12:18 | comment | added | Timothy Chow | Instead of "ZFC," maybe Tim Campion means the comprehension axiom, which gets off Russell's paradox on the technicality of being restricted. | |
| Jun 25, 2023 at 23:23 | comment | added | Joseph Van Name | I will leave it up to the reader and answerer whether they want to consider things such as Reinhardt cardinal as notable enough for us to be concerned about an inconsistency. | |
| Jun 25, 2023 at 23:22 | comment | added | Joseph Van Name | The rank-into-rank cardinals I0-I3 have a sophisticated theory behind them, so I am more interested in large cardinal axioms that follow from I0 cardinals than the exceedingly strong axioms. If Berkeley cardinals are found to be inconsistent, not much would change, but some axiom between I3 and I0 were found to be inconsistent, then that would be a big deal. I decided to narrow my question down to the parts of the large cardinal hierarchy that I (and probably others) care about, so I am not that interested in Berkeley cardinals at the moment. | |
| Jun 25, 2023 at 23:13 | comment | added | Joseph Van Name | I reformulated the question to try to make it more clear what I meant. Do you have a specific theorem proven in ZFC (rank-into-rank) that is close to being a contradiction in ZFC (rank-into-rank)? While the notion of a rank-into-rank cardinal may have originally be defined as a simple axiom that narrowly avoids the Kunen inconsistency, now we should know that the fact that if $j:V_\lambda\rightarrow V_\lambda$, then $\lambda$ has countable cofinality is essential to the theory of rank-into-rank embeddings since the quotient algebras of rank-into-rank embeddings must be locally finite. | |
| Jun 25, 2023 at 19:08 | history | answered | Tim Campion | CC BY-SA 4.0 |