Timeline for Can any poset of cardinality $\leq 2^{\aleph_0}$ be embedded in ${\cal P}(\omega)/(\text{fin})$?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 23, 2023 at 7:32 | vote | accept | Dominic van der Zypen | ||
| Aug 22, 2023 at 22:04 | answer | added | KP Hart | timeline score: 7 | |
| Aug 10, 2023 at 21:13 | comment | added | Ali Enayat | @An elaboration of Noah Schweber's comment: By a classical 1963 theorem of Parovicenko every Boolean algebra of cardinality $\aleph_1$ can be embedded into P(ω) mod FIN. | |
| Aug 9, 2023 at 12:21 | comment | added | Will Brian | @DominicvanderZypen: I think my answer to this question of yours answers this one as well: mathoverflow.net/questions/441234/… | |
| Aug 9, 2023 at 7:51 | comment | added | Dominic van der Zypen | @AndreasBlass thanks for your comment! I suspect it is not known whether the statement "${\frak c}$ cannot be embedded in ${\cal P}(\omega)/\text{(fin)}$" implies ${\sf (CH)}$? | |
| Aug 8, 2023 at 22:25 | comment | added | Joel David Hamkins | It might be nice to have a definitive summary account of the universality of $P(\omega)/\text{fin}$ posted as an answer. | |
| Aug 8, 2023 at 21:58 | comment | added | KP Hart | And Ken Kunen showed (implicitly) that after adding as many Cohen reals as you wish the ordinal $\omega_2+1$ is not a continuous image of $\omega^*$, hence its clopen algebra does not embed into $\mathcal{P}(\omega)/\mathrm{fin}$. | |
| Aug 8, 2023 at 21:47 | comment | added | Andreas Blass | @NoahSchweber's suspicion is correct. A negative answer is consistent with ZFC. In fact, the unembeddable poset $P$ can be just the ordinal $\mathfrak c$. This was proved by Peter Dordal in "A model in which the base matrix tree cannot have cofinal branches" (Journal of Symbolic Logic 52 (1987) pp. 651--664). | |
| Aug 8, 2023 at 20:27 | comment | added | Noah Schweber | This is true if $\mathsf{CH}$ holds; I suspect it's independent in general, but I'm not sure. | |
| Aug 8, 2023 at 20:23 | history | asked | Dominic van der Zypen | CC BY-SA 4.0 |