Timeline for What are the difficulties involved in proving that the Kunen inconsistency holds in $NGB$
Current License: CC BY-SA 4.0
12 events
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| Oct 23, 2018 at 17:09 | vote | accept | Thomas Benjamin | ||
| Oct 15, 2018 at 14:51 | history | edited | Thomas Benjamin | CC BY-SA 4.0 |
added words for completeness and a new tag
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| Oct 15, 2018 at 14:49 | answer | added | Gabe Goldberg | timeline score: 10 | |
| Oct 15, 2018 at 14:43 | comment | added | Thomas Benjamin | (cont.) or contrariwise, the existence (or possible nonexistence depending on the nature of the peculiarities) of the existence of Reinhardt cardinals in models of $NGB$ or $NGB$ + $Choice$? Finally, one could opt for an analysis of the gap in Rupert Mc'Callum's argument and how the result found in Laver's paper applies. I am not asking that the points given be necessary stumbling blocks--only that they may be possible stumbling blocks--and why (or why not) the possible stumbling blocks mentioned in an answer may be necessary. | |
| Oct 15, 2018 at 14:29 | comment | added | Thomas Benjamin | (cont.) Reinhardt cardinal, but then again there might be some structural peculiarities (mimicking the 'weirdness' of $ZF$ relative to $ZFC$ Asaf Karagila speaks of in his blog or his answers and comments in mathoverflow and mathstackexchange) of models $NGB$ relative to models of$NGBC$ or $NGB$ + $Choice$ (but if one could localize $Choice$ in $NGB$ + $Choice$ to somwhere above the critical point of an $I0$ embedding, and above that, $Choice$ would fail-- where would the point of demarcation be?), how would those peculiarities relate to the success or failure of the Kunen inconsistency | |
| Oct 15, 2018 at 14:06 | comment | added | Thomas Benjamin | @NoahSchweber: While one may not have a definitive answer, one might have relevant facts that could lead to a definitive answer. An easy example, there might be, say, in some obscure journal, a proof of the relevant Erdos-Hajnal theorem that doesn't use choice, or a choice principle weaker than $AC$ (say, the Axiom of Dependent Choices). The relevant Erdos-Hajnal theorem has been around a long time so such a theorem might be out there. A proof that the relevant Erdos-Hajnal theorem cannot do without $AC$ may suggest that one might be able to construct a model of $NGB$ that has a | |
| Oct 15, 2018 at 13:49 | comment | added | Thomas Benjamin | @AndreasBlass: Well, at least a proof of relative consistency.... | |
| Oct 13, 2018 at 23:26 | comment | added | Noah Schweber | To elaborate on Andreas' comment: since at present we don't know a definitive answer, we can't possibly show that a given point is a necessary stumbling block. The best that can be done is to point out the key step which "no known argument can get around," and observe that this step doesn't go through without choice. So I don't understand what you're looking for in an answer, short of a new major result in set theory, if that doesn't satisfy you. | |
| Oct 13, 2018 at 23:21 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
Added links, lo.logic tag
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| Oct 13, 2018 at 18:40 | review | Close votes | |||
| Oct 19, 2018 at 12:44 | |||||
| Oct 13, 2018 at 17:02 | comment | added | Andreas Blass | Although the title question is clear enough (and answered by the dependence of the proof on the Erdös-Hajnal theorem), the parenthetical addendum in the question itself, "must show how this fact allows for a non-trivial elementary embedding $j:V\to V$," seems to require a proof of consistency (relative to large cardinals) of NBG + existence of such $j$. | |
| Oct 13, 2018 at 16:19 | history | asked | Thomas Benjamin | CC BY-SA 4.0 |