For which class of forcings does the "name dichotomy" hold?

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?
• I probably should have added that the name dichotomy is easily provable for c.c.c. forcings. Jul 5, 2022 at 16:03
• Do you know a class of forcing for which it definitely does not hold? Jul 8, 2022 at 13:06
• Stationary set preserving forcings that are not semiproper, like Namba forcing in $L$, are counterexamples. I do not know any other unfortunately. Jul 8, 2022 at 15:03

1 Answer

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.