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A notion of forcing $P$ is called stationary set preserving iff each stationary subset of $\omega_1$ remains stationary in $V^P$. It is standard to show that semiproper (and of course proper) notions of forcing are stationary set preserving. On the other hand Shelah realized that assuming the Semiproper Forcing Axiom (which is consistent relative to a supercompact cardinal) each stationary set preserving notion is already semiproper. So the question naturally arises if there is a nice example of a notion of forcing which is stationary set preserving but not semiproper?

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up vote 6 down vote accepted

Namba forcing is stationary preserving but not semiproper unless Chang's Conjecture holds. See "Proper and Improper Forcing" of Shelah, Ch 12.

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