Timeline for Natural set-theoretic principles implying the Ground Axiom
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Oct 27, 2023 at 19:00 | comment | added | Joel David Hamkins | Ah, thanks for the explanation. | |
| Oct 27, 2023 at 17:39 | comment | added | Gabe Goldberg | @JoelDavidHamkins I was responding to Will Brian's question about principles that are always destroyed by set forcing. I guess my comment was a bit cryptic, but the idea was to come up with a principle that is always destroyed by set forcing but does not imply the Ground Axiom, so I did mean minimal in the sense you suggested. (Note that the mantle is the only definable element of the set-generic multiverse. But my example shows there are nontrivial definable subsets of the multiverse. Maybe the definability theory here is interesting...) | |
| Oct 27, 2023 at 15:07 | comment | added | Joel David Hamkins | @GabeGoldberg I would interpret the assertion "V is a minimal extension of the Mantle" as meaning that V is a minimal nontrivial extension of the Mantle, as in the usage of minimal Turing degrees or Sacks forcing giving minimal extensions. To get the ground axiom, you'd have to mean V is equal to the Mantle, which clearly just means there are no nontrivial grounds, i.e., the ground axiom. | |
| Oct 27, 2023 at 4:26 | vote | accept | Monroe Eskew | ||
| Oct 27, 2023 at 3:07 | comment | added | Joel David Hamkins | The first main theorems about it were proved by Jonas in his dissertation, including the fact that the CCA implies GA. Interestingly, he also proved that MA and PFA etc are consistent with the ground axiom. In particular, the common slogan that what those axioms assert is that a lot of forcing has already been done is not quite correct, if what is meant is set forcing. | |
| Oct 27, 2023 at 3:06 | comment | added | Joel David Hamkins | The ground axiom is due to Reitz and myself, as Jonas states in his paper. | |
| Oct 27, 2023 at 0:07 | history | edited | LSpice | CC BY-SA 4.0 |
Title of paper
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| Oct 26, 2023 at 20:51 | answer | added | Gabe Goldberg | timeline score: 12 | |
| Oct 25, 2023 at 20:00 | comment | added | Will Brian | @GabeGoldberg: I wouldn't count that one as obvious to me. I can't say whether it would be obvious to someone who's less ignorant of set-theoretic geology. | |
| Oct 25, 2023 at 19:35 | comment | added | Gabe Goldberg | @WillBrian Would you include something like "$V$ is a minimal extension of the mantle?" | |
| Oct 25, 2023 at 14:32 | comment | added | Kameryn Williams | This is perhaps a bit silly of an example, but axioms like $V = L$ or $V = L[0^\sharp]$ imply the ground axiom—the former because there are no (proper) inner models and the latter because $0^\sharp$ cannot be added by set forcing. | |
| Oct 25, 2023 at 14:27 | comment | added | Will Brian | I suppose any principle implying the Ground Axiom is destroyed by every set-sized forcing. Already I think it's an interesting question to ask what set-theoretic principles, other than the obvious ones, are always destroyed by set-sized forcing. (The "obvious" ones, to me, are things like $V=L$ or $V=L[\mu]$.) | |
| Oct 25, 2023 at 14:06 | history | asked | Monroe Eskew | CC BY-SA 4.0 |