My question(s) is the following:
- Is there a non-proper forcing notion which has the $\aleph_2$-cc property and doesn't collapse $\omega_1$?
- Is it consistent that such a forcing notion exists?
Ideas? Thanks
2026-03-30 08:23:56.1774859036
An $\aleph_2$-cc forcing which is not proper but doesn't collapse $\omega_1$
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It is certainly consistent, yes.
Assume $\sf CH$, and let $S\subseteq\omega_1$ be a stationary co-stationary set. Then the standard club shooting forcing, using closed subsets of $S$ ordered by end-extension, is not proper, since it kills the properness of $\omega_1\setminus S$, but by cardinality arguments alone it has size $\aleph_1$ so it has $\aleph_2$-c.c.
Note that this forcing is generally $S$-proper, so it might not be exactly what you're after.
Presumably you can modify this so that the club added is more like a Baumgartner club using finite conditions, which should be provably $\aleph_2$-c.c. in just $\sf ZFC$.