# What is the "Prikry–Silver collapse" when CH fails?

We all know and love Cohen reals, and we can (and often do) define the Cohen forcing as partial functions $$p\colon\omega\to 2$$ with finite domain. The Prikry–Silver forcing is defined as partial functions $$p\colon\omega\to 2$$ with co-infinite domain.

These two couldn't be any more different. For example, Cohen reals are aggressively non-minimal, whereas Prikry–Silver reals are minimal.

We can look at a similar situation with other forcings that are given by finite conditions. For example $$\operatorname{Col}(\omega,\omega_1)$$ is a forcing notion whose conditions are finite partial functions $$p\colon\omega\to\omega_1$$. We can ask what would be the Prikry–Silver analogue of this forcing, then. That is, $$\{p\colon\omega\to\omega_1\mid\operatorname{dom} p\text{ is co-infinite}\}$$.

Interestingly, assuming CH this is the same as the standard collapsing forcing. This follows from the fact that the cardinality of the partial order is $$2^{\aleph_0}$$, which under CH is just $$\aleph_1$$, and we know that any forcing of size $$\aleph_1$$ which collapses $$\omega_1$$ is equivalent to $$\operatorname{Col}(\omega,\omega_1)$$.

Question. Is this so-called "Prikry–Silver collapse" provably equivalent to the standard collapsing forcing?

• Prikry–Silver adds a minimal real, it is proper, and so it most certainly does not collapse $\omega_1$. May 20 at 14:56

If $$\mathrm{CH}$$ fails then $$\mathrm{Col}(\omega, \omega_1)$$ does not add a generic for the "Prikry-Silver collapse" $$\mathbb P$$: Let $$\mathbb U$$ be $$(\mathcal{P}(\omega)/I)^+$$ where $$I$$ is the ideal of finite subsets of $$\omega$$. The map $$\pi:\mathbb P\rightarrow \mathbb U,\ p\mapsto [\omega\setminus\mathrm{dom}(p)]_I$$ is a projection so that $$\mathbb P$$ adds a $$\mathbb U$$-generic filter. It is hence enough to show that under $$\neg\mathrm{CH}$$, $$\mathrm{Col}(\omega,\omega_1)$$ does not add a $$\mathbb U$$-generic filter. Suppose it does. In this case, there is a projection $$\mu:\mathrm{Col}(\omega,\omega_1)\rightarrow \mathrm{RO}(\mathbb U)$$ but this implies that there is $$p\in\mathbb U$$ and a set $$D\subseteq\mathrm{RO}(\mathbb U)$$ of size $$\omega_1$$ that is dense below $$p$$. This contradicts the fact that $$\mathbb U$$ has an antichain of size continuum below every condition.