Assume V=L. Is there a semi-proper notion of forcing that is not proper?


Namba forcing isn't semi-proper in L, and Prikry forcing isn't even available there.



Semiproper but not proper
The following is proved as Claim 2.1 in Shelah's paper ``[Forcing axiom failure for any $\lambda>\aleph\_1$](http://link.springer.com/article/10.1007%2Fs00153-003-0208-9)''



> 
> **Theorem.** There is a forcing notion of size $2^{\aleph\_2}$ which is not proper but
>  is $\{\aleph\_1\}$-semi proper.
> 
> 
> 


In fact Shelah proves more: for every regular $κ>ℵ\_1$ there is a forcing notion $P$ of cardinality $2^κ$ such that there is a stationary set $S⊆[2^κ]^{ℵ\_0}$ not preserved by $P$ and if $χ>2^κ$, $p∈P∈N≺(H(χ),∈), N$ countable, then there is $q∈P$ stronger than $p$ and $q$ forces that $N∩κ$ is an initial segment of $N[G]∩κ.$ 


