Timeline for Strong chains of uncountable functions and cardinal characteristics
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 4, 2020 at 14:21 | comment | added | Todd Eisworth | Yes. And once you have a long strong chain, it will still exist in further ccc extensions, so you can achieve further configurations, say by adding a small dominating family. | |
| Mar 4, 2020 at 5:37 | comment | added | Jing Zhang | It seems that the forcing to add such an $\omega_2$ strong chain is strongly proper, so at least we can exclude invariants like $\mathfrak{b}$ (and many others) since no strongly proper forcing can add a dominating real (any real lives in a Cohen sub-extension but Cohen forcing can't add a dominating real). | |
| Mar 3, 2020 at 21:00 | comment | added | Todd Eisworth | I don't the answer to the $\omega_3$ question. What I'm really curious about is if the existence of the strong $\omega_2$-chain implies that some cardinal characteristics must be greater than $\omega_1$ (we know that $\mathfrak{c}>\omega_1$, for example). | |
| Mar 3, 2020 at 16:09 | comment | added | Jing Zhang | so Chang's Conjecture keeps this $\kappa$ small ($\aleph_1$) and CC is c.c.c indestructible so basically this $\kappa$ cannot bound any cardinal invariant. For the other direction, is it known if it is possible to have $\kappa\geq \omega_3$? | |
| Mar 1, 2020 at 22:40 | history | asked | Todd Eisworth | CC BY-SA 4.0 |