Timeline for PFA for cardinal preserving forcing notions and the CH
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Oct 30, 2020 at 16:13 | comment | added | Todd Eisworth | This is related to Rahman Muhammadpour's question on whether forcing axioms for proper posets of cardinality $\aleph_1$ imply the continuum is $\aleph_2$, See: mathoverflow.net/questions/350880/… | |
| Oct 28, 2020 at 19:23 | comment | added | Asaf Karagila♦ | If it determines the value, it has to be $\aleph_2$. So I guess your question can be clarified by asking specifically if this forcing axiom is consistent with a larger continuum. This is a question I'd ask David Aspero, not MathOverflow, to be honest. I'll alert him to this. | |
| Oct 28, 2020 at 15:41 | comment | added | Mohammad Golshani | I mean determine its value, like what PFA does, namely it implies the continuum is $\aleph_2$. | |
| Oct 28, 2020 at 15:14 | comment | added | Todd Eisworth | By "decide CH", do you mean just whether CH is true or false? Because hitting $\aleph_1$ dense subsets of Cohen forcing implies that CH is false, right? But deciding the value of the continuum is another matter. | |
| Oct 28, 2020 at 14:16 | comment | added | Asaf Karagila♦ | Also, note that this class of forcings has a terrible iteration properties. | |
| Oct 28, 2020 at 14:11 | comment | added | Asaf Karagila♦ | Are you married to the idea that we can only intersect $\aleph_1$ dense open sets? Normally to get a large continuum (as a consequence) we need to intersect more. | |
| Oct 28, 2020 at 14:10 | comment | added | Mohammad Golshani | @AsafKaragila. Yes, sure, i mean given any $\kappa,$ can we get it with the continuum above $\kappa$?. Th motivation comes from the fact that to show PFA decides CH, we use forcings which collapse the continuum!. | |
| Oct 28, 2020 at 13:23 | comment | added | Asaf Karagila♦ | If mean, it implies MA (because every ccc forcing is proper and cardinal preserving), so presumably it decides $\lnot\sf CH$. Since it follows from $\sf PFA$, it's compatible with $2^{\aleph_0}=\aleph_2$. So by "large" do you mean larger than $\aleph_2$? | |
| Oct 28, 2020 at 10:39 | history | edited | Martin Sleziak |
added (continuum-hypothesis) - to me it seems as a suitable tag, feel free to revert the edit if it does not fit for some reason
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| Oct 28, 2020 at 10:36 | history | asked | Mohammad Golshani | CC BY-SA 4.0 |