Timeline for Existence of inner models of $\mathrm{ZFC} \ +$ forcing axioms, under incompatible assumptions
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 5, 2021 at 20:43 | comment | added | Grigor | it seems you could just use Sigma^1_3 absoluteness and get an inner model of ZFC+MM^++ in L[mu], no? repeat 0^# argument but with a measurable, like Gabe is doing below, iterability is Pi^1_2, so you don't need Woodins, just Sigma^1_3 abs. | |
| Apr 1, 2021 at 16:29 | comment | added | Monroe Eskew | @Zoorado This is a current conjecture as far as I understand. Namely that some strong hypothesis should imply that the HOD of a canonical inner model of AD (larger than $L(\mathbb R)$) should contain large enough cardinals to force PFA. | |
| Apr 1, 2021 at 11:01 | comment | added | Zoorado | Yes, you are right. I was mainly curious if there were proper extensions of ZF+AD which can give rise to inner models of ZFC+PFA. | |
| Apr 1, 2021 at 10:36 | comment | added | Asaf Karagila♦ | @Zoorado: PFA implies $\sf AD$ in $L(\Bbb R)$, and I seem to recall that it implies even more than just that. If that is indeed the case, then Con(ZFC+PFA) is strictly greater than Con(ZF+AD). | |
| Apr 1, 2021 at 9:46 | comment | added | Monroe Eskew | @Zoorado Yes, that sounds right. | |
| Apr 1, 2021 at 9:42 | comment | added | Zoorado | Am I right to say that, if I replace $\mathrm{MA}$ with $\mathrm{PFA}$ in the first question, the answer is also unknown? | |
| Apr 1, 2021 at 9:41 | vote | accept | Zoorado | ||
| Apr 1, 2021 at 9:37 | history | answered | Monroe Eskew | CC BY-SA 4.0 |