Let $\mathbb P$ be a forcing that does not collapse $\omega\_1$, $\theta$ sufficiently large and regular and $X\prec H\_\theta$ a countable elementary substructure with $\mathbb P\in X$ as well as $p\in \mathbb P\cap X$. This is a typical situation if one deals with (semi)proper forcings. Let $\delta=X\cap\omega\_1$. Assume further that $\dot S\in X$ is a $\mathbb P$-name for a subset of $\omega\_1$. I would like to find a $(X, \mathbb P)$-semigeneric condition $q\leq p$ and simultaneously control whether $\delta\in\dot S^G$ or not. Lets say we ask for such $q$ so that
$$q\Vdash \check\delta\notin\dot S$$
There are obvious obstructions to this: If there is $T\in \mathcal P(\omega\_1)\cap X$ with $\delta\in T$ and
$$p\Vdash\check T\subseteq \dot S \mod \mathrm{NS}\_{\omega\_1}$$
then there cannot be such $q$. I can prove that if the Strong Reflection Principle ($\mathrm{SRP}$) holds, this is the "only reason" for the nonexistence of such $q$:



> 
> Assume $\mathrm{SRP}$ holds. Let $\mathbb P, X, p, \dot S$ as above. Exactly one of the following holds:
> 
> 
> 1. There is $q\leq p$ that is $(\mathbb P, X)$-semigeneric and $q\Vdash\check\delta\notin \dot S$
> 2. There is $T\in \mathcal P(\omega\_1)\cap X$ with $\delta\in T$ and $$p\Vdash\check T\subseteq \dot S \mod \mathrm{NS}\_{\omega\_1}$$
> 
> 
> 


The proof of $\neg 1.\Rightarrow 2.$ is similar to the the proof of "stationary set preserving forcings are semiproper" (under $\mathrm{SRP}$). Let us say the name dichotomy holds for $\mathbb P$ if exactly one of 1. and 2. holds for any $X, p, \dot S$ as above. One can hope that if a property is provable for the class of $\omega\_1$-preserving forcings under $\mathrm{SRP}$ then maybe this property holds in general for more "well-behaved" classes of forcings. So my question is:



> 
> Is it provable (in $\mathrm{ZFC}$) that the name dichotomy holds for
> 
> 
> * $\sigma$-closed forcings?
> * proper forcings?
> * semiproper forcings?
> 
> 
> 



For which class of forcings does the "name dichotomy" hold?
It turns out that it is not provable in $\mathrm{ZFC}$ that even $\sigma$-closed forcings satisfy the name dichotomy. The answer to all three questions is thus no (unfortunately).


I will show that under $V=L$, $\mathbb P:=\mathrm{Add}(\omega\_1, 1)$ does not satisfy the name dichotomy:


Consider the function $f:\omega\_1\rightarrow \omega\_1$ defined via $f(\alpha)$ is the least $\beta$ so that $\alpha$ is countable in $L\_{\beta+1}$ (this is the wellknown example of a function in $L$ not bounded by any canonical function).
If $G$ is $\mathbb P$-generic, then in $V$ we can define the following set:
$$S=\{\alpha<\omega\_1\mid G\upharpoonright\alpha\text{ is generic over }L\_{f(\alpha)}\}$$
Here, "$G\upharpoonright \alpha$ is generic over $L\_{f(\alpha)}$" means that $G\upharpoonright \alpha=\{p\in G\mid \mathrm{dom}(p)\subseteq\alpha\}$ is generic for $\mathrm{Add}(\alpha, 1)^{L\_{f(\alpha)}}$ over $L\_{f(\alpha)}$ (in particular $G\upharpoonright\alpha\subseteq L\_{f(\alpha)}$). Let $\dot S$ be a name in $V$ for this set $S$.



> 
> Claim: $S$ is stationary costationary in $V[G]$.
> 
> 
> 


Proof: It is not difficult to see that $S$ is stationary. To see that $S$ is costationary, let $\dot C$ be a name for a club in $V$. Let $\theta$ be large enough, regular and $X\prec H\_\theta$ countable with $\dot C\in X$. There is a descending sequence $\vec p:=\langle p\_\alpha\mid \alpha<\omega\_1\rangle\in H\_\theta$ of conditions in $\mathbb P$ so that for all $\alpha$ there is $\alpha\leq\beta$ so that $p\_\alpha\Vdash\check\beta\in \dot C$. By elementarity, we may assume $\vec p\in X$. Let $q=\bigcup\_{\alpha<\delta^X} p\_\alpha$. It is clear that $q$ is *not* $(X,\mathbb P)$-generic, but we have $q\Vdash\check \delta^X\in \dot C$. It follows that $q\Vdash\check\delta^X\in\dot C-\dot S$. $\square$


The argument above shows even a little more: If $T\in V$ is stationary in $\omega\_1$, then $T\cap S$, $T-S$ are stationary in $V[G]$ (just choose $X$ with $\delta^X\in T$). It follows that 2. of the name dichotomy fails for $\dot S$ (for any appropriate $\theta$, $X$, $p$).


To get a failure of 1. as well, it is enough to find a countable $X\prec H\_\theta$ so that if $X\cong L\_\gamma$ then $\mathcal P(\delta)\cap L\_\gamma=\mathcal P(\delta)\cap L\_{f(\delta)}$ for $\delta=\delta^X$: Any $q$ that is $(X, \mathbb P)$-semigenric is actually $(X, \mathbb P)$-generic (as $\mathbb P$ has size $\omega\_1$) and hence as $L\_\gamma$ and $L\_{f(\delta)}$ have the same dense subsets of $\mathrm{Add}(\delta, 1)$, such $q$ forces $G\upharpoonright \delta$ to be generic over $L\_{f(\delta)}$.


The following example of such an $X$ is due to Ralf Schindler:


Let $X\_0=\mathrm{Hull}^{H\_\theta}(\emptyset)$ and for $n<\omega$ let $X\_{n+1}=\mathrm{Hull}^{H\_\theta}(\{X\_n\})$. Put $X=\bigcup\_{n<\omega} X\_n$. I claim that this $X$ works. Let $\delta=\delta^X$. Let $X\cong L\_\gamma$.



> 
> Claim: $f(\delta)=\gamma+1$.
> 
> 
> 


Proof: $L\_\gamma$ is a model of $\mathrm{ZF}^-$, $\delta$ is uncountable in $L\_\gamma$ and so $f(\delta)>\gamma$. For $n<\omega$ let $X\_n\cong L\_{\gamma\_n}$. Then $\langle \gamma\_n\mid n<\omega\rangle$ is definable over $L\_{\gamma+1}$: $\gamma\_0$ is the least $\xi$ with $L\_\xi\prec L\_{\gamma}$ and $\gamma\_{n+1}$ is the least $\gamma\_n<\xi$ with $L\_\xi\prec L\_{\gamma}$. Thus $\langle \delta^{X\_n}\mid n<\omega\rangle$ is definable over $L\_{\gamma+1}$ and as $\delta=\mathrm{sup}\_{n<\omega} \delta^{X\_n}$, $\delta$ is countable in $L\_{\gamma+2}$. $\square$


As $L\_\gamma$ is a model of $\mathrm{ZF}^{-}$ we have
$$\mathcal P(\delta)\cap L\_\gamma=\mathcal P(\delta)\cap L\_{\gamma+1}$$
so that $X$ has the desired property.


