Timeline for "Antiforcing" - Is there a method to 'remove' sets from a model of ZF?
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| Sep 28, 2018 at 20:10 | comment | added | Keith Millar | I see what you meant. I think that this would show that a specific large cardinals with NO known inner models, one could "antiforce" into a model in which $|(\kappa^+)^M|=\kappa^+$ for a given inner model $M$. | |
| Sep 28, 2018 at 6:22 | comment | added | Ralf Schindler | ?? $|(\kappa^+)^K| \not= \kappa^+$ is equivalent with $|(\kappa^+)^K|=\kappa$. | |
| Sep 27, 2018 at 21:30 | history | edited | Keith Millar | CC BY-SA 4.0 |
core models dont work that way
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| Sep 27, 2018 at 21:29 | comment | added | Keith Millar | Oh, that's correct. I should have said "in which $|(\kappa^+)^K|\neq\kappa^+$." | |
| Sep 27, 2018 at 11:45 | review | Close votes | |||
| Oct 3, 2018 at 10:10 | |||||
| Sep 27, 2018 at 8:59 | comment | added | Ralf Schindler | Your question is too vague to say much more than what Andrés already did by way of an answer. Concerning your item 3, though, which large cardinals $\kappa$ do you have in mind? $|(\kappa^{+})^K| = \kappa$ is plain false if $\kappa$ is weakly compact (or e.g. carries a precipitous ideal). | |
| Sep 25, 2018 at 23:09 | comment | added | Keith Millar | @AsafKaragila I suppose so, if it would make it easier to read. It just helps me to think of it like it already exists so I can surmise some properties it would have if it were to exist. I see your point though. | |
| Sep 25, 2018 at 6:23 | comment | added | Asaf Karagila♦ | Can you do me a small favor, and stop giving weird names to everything? Especially if you ask "Hey, did someone think about this concept yet". | |
| Sep 24, 2018 at 22:40 | comment | added | Keith Millar | @AndrésE.Caicedo I see. Thank you! This is a quite interesting subject. I believe you are right that this is exactly what I am after. | |
| Sep 24, 2018 at 20:06 | comment | added | Andrés E. Caicedo | Keith, I didn't just say inner model theory, but rather inner model-theoretic geology. In any case, one of the key motivators of inner model theory is very close to what you are after. The evolution of covering lemmas illustrates this dramatically. | |
| Sep 24, 2018 at 19:06 | comment | added | James E Hanson | @KeithMillar Assuming we're talking about inner models with the same ordinals, shouldn't it be the case that it's possible to remove some set $A$ while preserving some set $B$ if and only if $A \notin L(B)$? | |
| Sep 24, 2018 at 18:58 | history | edited | Keith Millar | CC BY-SA 4.0 |
added more question
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| Sep 24, 2018 at 18:56 | comment | added | Keith Millar | @YCor this is true; but not all sets can be removed (in particular none of the constructible sets can be removed). Also, as far as I know, removing sets but preserving others is a bit more difficult, which is what "antiforcing" would be used for. Of course, not being able to remove constructible sets isn't a problem because any inner model contains all of the constructible sets anyways. | |
| Sep 24, 2018 at 18:54 | comment | added | Keith Millar | I understand what inner model theory is; I meant a sort of different method which is standardized (there are only a couple known inner models, meanwhile there are many known forcing universes). Specifically, I would like to know if there is a method of making inner-model-like things similar to "undoing" forcing. | |
| Sep 24, 2018 at 18:52 | comment | added | Andrés E. Caicedo | You should probably read about set-theoretic geology and its inner model-theoretic version. | |
| Sep 24, 2018 at 18:49 | comment | added | YCor | I may misunderstand, but "removing" sets was initially considered as easier, leading to proofs of the consistency of AC or CH (assuming that of ZF), much before Cohen introduced forcing to prove consistency of their negation. | |
| Sep 24, 2018 at 18:47 | history | asked | Keith Millar | CC BY-SA 4.0 |