It is easy to see a forcing of size $\aleph_1$ is proper if and only if is semiproper. I was wondering when such an equivalency holds between semi-proper and stationary-preserving forcings in $\rm ZFC$? Or consistently in a model where significant fragments of $\rm MM$ fail.
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3$\begingroup$ If you replace $\aleph_1$ by $\aleph_0$, then there is a simple answer. $\endgroup$– Asaf Karagila ♦Commented Dec 13, 2020 at 9:53
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$\begingroup$ Yep, that universal answer! you can even prove more $\endgroup$– Rahman. MCommented Dec 13, 2020 at 10:12
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3$\begingroup$ Maybe related: Collapsing ω2 with semi-proper forcing $\endgroup$– Mohammad GolshaniCommented Dec 13, 2020 at 10:43
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1 Answer
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The answer is no, as the following upcoming work of Shelah and Usuba shows:
Theorem (Shelah-Usuba): The following theories are equiconsistent with ZFC:
ZFC+CH+ “there is an $\omega _1$-stationary preserving $\sigma$-Baire poset of size $\aleph_1$ which is not semiproper”.
ZFC+“Martin’s axiom for semiproper posets of size $\aleph_1$” + “there is an $\omega _1$-stationary preserving $\sigma$-Baire poset of size $\aleph_1$ which is not semiproper”.
ZFC+CH+“every $\omega _1$-stationary preserving $\sigma$-Baire poset of size $\aleph_1$ is semiproper”.
See $\omega_1$-Stationary preserving $\sigma$-Baire posets of size $\aleph_1$.
answered Mar 12, 2021 at 12:20