Timeline for Why not adopt the constructibility axiom $V=L$?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| May 24, 2019 at 6:12 | comment | added | Alec Rhea | @JoelDavidHamkins Just saw this, very cool once again and thank you. | |
| May 21, 2019 at 6:43 | comment | added | Joel David Hamkins | Yes, there can be countable transitive models of Kelley-Morse set theory. If there is any transitive model at all (and this follows from an inaccessible cardinal), then there is a countable one by taking an elementary substructure and the Mostwski collapse. | |
| May 21, 2019 at 3:17 | comment | added | Alec Rhea | Very cool answer -- I'm sure this is an elementary question, but do set theories 'like $MK$' which can deal with proper classes as their subsets have countable (transitive) models? | |
| May 20, 2019 at 15:32 | comment | added | Joel David Hamkins | Meanwhile, yes, the theorems are known to fail (consistently) for uncountable models. For example, in my paper jdh.hamkins.org/…, we show that there are incomparable $\omega_1$-like models of set theory, which violate the comparability property of the theorem I had stated for countable models. | |
| May 20, 2019 at 15:29 | comment | added | Joel David Hamkins | My argument concerning the significance of the results for $V=L$ is that they undermine the "restrictive" aspect of $V=L$, which is the main mark against it. The results explain how we can have large cardinals in one universe, and $V=L$ in a larger universe, on and off again as the ordinals arrive. But yes, these facts are proved in ZFC only for countable models, which is viewed as a "toy" multiverse. We have in principle no way to prove things like this for the full actual multiverse, except in analogy with the toy multiverse. | |
| May 20, 2019 at 15:19 | comment | added | Andrej Bauer | "Needs" as in "it is known to be false for uncountable models"? I am asking because I am not sure how to understand the significance of the theorems for the question at hand (should we adopt $V = L$?). | |
| May 20, 2019 at 14:38 | comment | added | Joel David Hamkins | If you are referring to Barwise's theorem or mine, then yes, one needs the restriction to countable models in order to prove the theorems. | |
| May 20, 2019 at 14:35 | comment | added | Andrej Bauer | Is the restriction to countable models a technical convenience, a necessity, or something else? | |
| S May 20, 2019 at 14:22 | history | answered | Joel David Hamkins | CC BY-SA 4.0 | |
| S May 20, 2019 at 14:22 | history | made wiki | Post Made Community Wiki by Joel David Hamkins |