Timeline for Formulating Kunen's inconsistency and Reinhardt cardinals in term of category theory
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| May 11, 2015 at 5:47 | comment | added | Asaf Karagila♦ | @Andreas: Now I feel less guilty for not thinking about $\sf KM$ as something with global choice included. :-) | |
| May 11, 2015 at 5:19 | answer | added | Tim Campion | timeline score: 6 | |
| May 10, 2015 at 23:58 | comment | added | Joel David Hamkins | That's funny! There may be less standardization of this than I had thought. But I think it will be helpful for us all to push for standardized usage. The issue of global choice in GBC is often discussed, since it is an issue for the conservativity result over ZFC, which would be completely trivial without the "global" aspect, since one could just take definable classes. | |
| May 10, 2015 at 23:24 | comment | added | Andreas Blass | @JoelDavidHamkins Thanks for the clarification. If I remember correctly, global choice was in both Kelley's and Morse's axiomatizations, but it was also in Gödel's version of GB (Axiom E, I think); I'm not sure about Bernays, but von Neumann had it in the form "all proper classes are in bijection with V". So it seems the presence of choice in a system named after someone need not agree with its presence in that person's axioms. (That's certainly the case for Zermelo and ZF.) | |
| May 10, 2015 at 23:13 | comment | added | Joel David Hamkins | @AndreasBlass Yes, of course the difference you mention is the most important difference. But my understanding of the conventional usage is that when one says Kelley-Morse (KM), one usually means to include global choice, but when one says Goedel-Bernays (GB or NBG), one does not. Goedel-Bernays with choice (usually written GBC or NGBC) means with global choice. After Googling, however, I see that there may not be universal agreement on whether global choice is part of KM, although it does seem to be usually included and it also seems to be part of the original theory in Kelley's book. | |
| May 10, 2015 at 22:56 | comment | added | Andreas Blass | @JoelDavidHamkins I never realized that KM and GB differ in regard to global choice. I thought the difference between them was just whether the class comprehension axioms allow quantification of class variables, and that the presence or absence of global choice was an orthogonal issue. | |
| May 10, 2015 at 16:15 | comment | added | Joel David Hamkins | Well, KM includes global choice, and so it refutes Reinhardt cardinals... | |
| May 10, 2015 at 15:55 | comment | added | Asaf Karagila♦ | @Joel: Kinda surprised to see you omitted $\sf KM$! :-) | |
| May 10, 2015 at 12:36 | comment | added | Joel David Hamkins | It is not really correct to talk about Reinhardt cardinals in mere ZF, since the concept is not first-order expressible in the language of set theory, unless it is self-contradictory (proof: if being Reinhardt were first-order expressible and $\kappa$ is the least Reinhardt cardinal, with elementary embedding $j:V\to V$, then $j(\kappa)$ is also the least Reinhardt cardinal, a contradiction). Rather, formalizing Reinhardt cardinals requires some formal treatment of classes, and so one is working in GB or ZF($j$). | |
| May 10, 2015 at 11:52 | comment | added | Asaf Karagila♦ | @David: I merely wanted to point out that if we are referring to possible cases that elementary embeddings and ultrapowers are not quite as nicely behaved as in $\sf ZFC$, these two papers should be mentioned. :-) | |
| May 10, 2015 at 11:48 | comment | added | David Roberts♦ | @Asaf the example I linked to is obviously the smallest large cardinal described in this way. I'd be very interested to see a topos version of Reinhardt, in the fully intuitionistic, choice-free setting. | |
| May 10, 2015 at 10:05 | history | edited | Asaf Karagila♦ | CC BY-SA 3.0 |
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| May 10, 2015 at 10:04 | comment | added | Asaf Karagila♦ | @David: One link and another link are probably worth a mention here. | |
| S May 10, 2015 at 10:02 | history | suggested | Rahman. M |
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| May 10, 2015 at 9:51 | review | Suggested edits | |||
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| May 10, 2015 at 9:36 | comment | added | David Roberts♦ | I've been thinking lately it would be a good idea to link elementary embeddings and category theoretic constructions as discussed at eg golem.ph.utexas.edu/category/2015/04/five_quickies.html#c048923. One issue to note is that measurables and certain elementary embeddings aren't the same in the absence of Choice, and this makes things more interesting. | |
| May 10, 2015 at 8:34 | history | asked | Thomas Benjamin | CC BY-SA 3.0 |